#6606532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1771 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)(19,34)(20,26)(21,33)(25,40)(27,39)(28,36)(30,44)(31,43)(32,38)(35,41)(37,42);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,26)(14,20,23,24)(16,30,31,25)(17,27,19,28)(21,37,38,35)(22,36,34,39)(29,40,43,44)(32,42,33,41);(6,9)(7,11)(13,16,17,32,18,31,19,33)(14,21,34,43,23,38,22,29)(15,25,27,41,26,30,28,42)(20,35,39,40,24,37,36,44);(5,7)(6,10)(8,11)(9,12)(15,26)(16,31)(20,24)(21,38)(27,28)(29,43)(32,33)(36,39);(6,9)(7,11)(13,17,18,19)(14,22,23,34)(15,27,26,28)(16,32,31,33)(20,36,24,39)(21,29,38,43)(25,41,30,42)(35,44,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513728);