#478650

order	512
nilpotency class	3
isoclinism	isoclinic class of #476143
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2819
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17159
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

783 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385846);