#6443260

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
prime radical	C₂²×#576	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5086 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)(14,31)(15,20)(16,23)(17,22)(19,26)(24,38)(27,34)(28,33)(29,32)(30,37)(35,39)(36,40);(1,2,3,4)(5,6)(7,8)(9,11,14,23,15,24,26,27)(10,16,19,21,20,34,31,38)(12,13,22,30,28,29,39,40)(17,32,25,37,35,18,33,36);(6,8)(9,12,26,39,15,28,14,22)(10,17,31,33,20,35,19,25)(11,13,27,40,24,29,23,30)(16,32,38,36,34,18,21,37);(6,8)(9,13)(10,18)(11,22)(12,27)(14,30)(15,29)(16,33)(17,21)(19,36)(20,32)(23,28)(24,39)(25,34)(26,40)(31,37)(35,38);(12,28)(13,29)(17,35)(18,32)(22,39)(25,33)(30,40)(36,37)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,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, 6350456);