#6465645

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3240 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)(9,10,14,16,15,20,13,19)(11,17,21,28,23,30,24,32)(12,18,25,29,27,33,26,34)(22,31,35,38,36,39,37,40);(2,4)(10,16)(11,21)(12,25)(13,14)(17,30)(18,33)(19,20)(22,36)(23,24)(26,27)(31,40)(38,39);(5,6,7,8)(9,11,14,21,15,23,13,24)(10,17,16,28,20,30,19,32)(12,22,25,35,27,36,26,37)(18,31,29,38,33,39,34,40);(9,12,13,26,15,27,14,25)(10,18,19,34,20,33,16,29)(11,22,24,37,23,36,21,35)(17,31,32,40,30,39,28,38);(11,23)(17,30)(21,24)(22,36)(28,32)(31,39)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=hi,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^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, 6372841);