#6443259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

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, 6350455);