#6814796

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#29736	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,17,19)(15,23,24,39)(16,27,28,21)(18,32,31,22)(20,34,33,26)(25,40,29,38)(30,37,41,43)(35,42,36,44);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,24)(16,28)(22,32)(26,34)(30,41)(35,36)(38,40);(1,2,4,3)(13,15,18,26,17,24,31,34)(14,20,22,23,19,33,32,39)(16,29,30,44,28,25,41,42)(21,36,37,40,27,35,43,38);(6,10)(9,11)(13,16,17,28)(14,21,19,27)(15,25,24,29)(18,30,31,41)(20,35,33,36)(22,37,32,43)(23,38,39,40)(26,42,34,44);(1,3,4,2)(7,12)(9,11)(15,24)(16,28)(20,33)(21,27)(23,39)(26,34)(30,41)(37,43)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721992);