#6668871

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2085 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)(13,14,19,18)(15,20,25,31)(16,21,27,32)(17,22,30,33)(23,34,37,26)(24,35,39,29)(28,36,41,42)(38,43,44,40);(6,9)(7,11)(13,15,19,25)(14,20,18,31)(16,26,27,34)(17,29,30,35)(21,23,32,37)(22,24,33,39)(28,38,41,44)(36,43,42,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,28)(18,32)(19,27)(20,34)(22,36)(24,38)(25,37)(26,31)(29,40)(30,41)(33,42)(35,43)(39,44);(6,9)(10,12)(16,27)(21,32)(23,37)(26,34)(28,41)(36,42)(38,44)(40,43);(13,17,19,30)(14,22,18,33)(15,24,25,39)(16,28,27,41)(20,35,31,29)(21,36,32,42)(23,38,37,44)(26,40,34,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6576067);