#6814778

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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×#3204	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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)(13,14,19,23)(15,20,26,36)(16,27,30,39)(17,22,18,33)(21,32,40,29)(24,41,28,44)(25,35,31,37)(34,42,38,43);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,23,36)(16,28,30,24)(17,31,18,25)(21,38,40,34)(22,37,33,35)(27,44,39,41)(29,42,32,43);(1,2,4,3)(13,16,18,29,19,30,17,32)(14,21,22,27,23,40,33,39)(15,24,25,43,26,28,31,42)(20,34,37,41,36,38,35,44);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,25,26,31)(16,29,30,32)(20,35,36,37)(21,39,40,27)(24,42,28,43)(34,41,38,44);(1,3,4,2)(7,12)(9,11)(15,26)(16,30)(20,36)(21,40)(25,31)(27,39)(29,32)(35,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721974);