#391640

order	512
nilpotency class	3
isoclinism	isoclinic class of #391639
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14649
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

689 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,12,7,11,23,34)(3,8,16,27,15,26,42,51)(4,9,19,30,18,29,22,35)(5,10,21,33,20,32,46,58)(13,36,39,52,38,24,43,49)(14,25,41,50,40,31,56,57)(17,28,45,55,44,54,47,59)(37,53,60,64,61,48,62,63);(1,3)(2,8)(4,13,18,38)(5,14,20,40)(6,16)(7,15)(9,24,29,36)(10,31,32,25)(11,26)(12,27)(17,37)(19,39,22,43)(21,41,46,56)(23,42)(28,53)(30,49,35,52)(33,57,58,50)(34,51)(44,61)(45,60)(47,62)(48,54)(55,64)(59,63);(1,4)(2,9,11,29)(3,13,15,38)(5,17,20,44)(6,22)(7,18)(8,24)(10,28)(12,35,34,30)(14,37)(16,43,42,39)(19,23)(21,47,46,45)(25,48,31,53)(26,36)(27,52)(32,54)(33,59)(40,61)(41,62)(49,51)(50,64,57,63)(55,58)(56,60);(1,5)(2,10)(3,14,15,40)(4,17,18,44)(6,21)(7,20)(8,25,26,31)(9,28,29,54)(11,32)(12,33)(13,37)(16,41,42,56)(19,45,22,47)(23,46)(24,48)(27,50,51,57)(30,55,35,59)(34,58)(36,53)(38,61)(39,60)(43,62)(49,63)(52,64)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=g,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286106, #72286126
maximal normal subgroup of	#72284899, #72286106, #72286126

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298836);