#6443267

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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
prime radical	C₂²×#577	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4978 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)(12,25)(13,18)(14,30)(15,20)(16,23)(17,27)(19,31)(22,35)(24,34)(26,36)(28,32)(29,38)(33,40)(37,39);(1,2,3,4)(5,6)(7,8)(9,11,14,23,15,24,31,40)(10,16,19,21,20,33,30,34)(12,13,27,29,26,28,22,39)(17,32,25,38,35,18,36,37);(6,8)(9,12,14,27,15,26,31,22)(10,17,19,25,20,35,30,36)(11,13,23,29,24,28,40,39)(16,32,21,38,33,18,34,37);(6,8)(9,13)(10,18)(11,22)(12,23)(14,29)(15,28)(16,25)(17,34)(19,37)(20,32)(21,35)(24,27)(26,40)(30,38)(31,39)(33,36);(12,26)(13,28)(17,35)(18,32)(22,27)(25,36)(29,39)(37,38)

Polycyclic presentation:

2^9|b^2=f,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350463);