#6682541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589737);