#6467693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

1862 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:

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

Polycyclic presentation:

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

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, 6374889);