#6668649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575845);