#6465561

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3238 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)(9,10,13,17,14,18,27,34)(11,19,22,29,21,33,26,15)(12,16,25,32,24,31,39,40)(20,35,36,30,23,37,38,28);(6,7)(9,11)(10,15)(12,23)(13,26)(14,21)(16,30)(17,33)(18,29)(19,34)(20,24)(22,27)(25,36)(28,31)(32,35)(37,40)(38,39);(1,2,3,4)(5,6,8,7)(9,12,14,24)(10,16,18,31)(11,20,21,23)(13,25,27,39)(15,28,29,30)(17,32,34,40)(19,35,33,37)(22,36,26,38);(5,7,8,6)(11,21)(15,29)(19,33)(20,23)(22,26)(28,30)(35,37)(36,38);(12,24)(16,31)(20,23)(25,39)(28,30)(32,40)(35,37)(36,38)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6372757);