#6465734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3188 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372930);