#6630304

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2124 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,19,24,32)(16,26,28,41)(17,30,27,21)(20,34,35,29)(23,36,38,44)(25,40,37,33)(31,42,43,39);(6,9)(10,12)(13,15,18,24)(14,19,22,32)(16,23,28,38)(17,25,27,37)(20,31,35,43)(21,33,30,40)(26,36,41,44)(29,39,34,42);(13,16,27,34,18,28,17,29)(14,20,30,41,22,35,21,26)(15,23,37,42,24,38,25,39)(19,31,40,44,32,43,33,36);(2,4)(15,24)(16,28)(19,32)(20,35)(25,37)(26,41)(29,34)(33,40);(5,7,8,11)(6,10,9,12)(15,24)(19,32)(23,38)(25,37)(31,43)(33,40)(36,44)(39,42)

Polycyclic presentation:

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

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, 6537500);