#6606436

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513632);