#6816334

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,e^2=fh,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, 6723530);