#477010

order	512
nilpotency class	3
isoclinism	isoclinic class of #477009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17102
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1563 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1563 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286090, #72286126, #72286231
maximal normal subgroup of	#72286090, #72286126, #72286231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384206);