#6630343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ci,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537539);