#6465583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

3196 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)(9,10,14,18,15,19,29,35)(11,20,23,31,24,34,28,16)(12,25,26,40,13,17,27,33)(21,37,38,30,22,36,39,32);(6,7)(9,11,15,24)(10,16,19,31)(12,22,13,21)(14,28,29,23)(17,32,25,30)(18,34,35,20)(26,38,27,39)(33,36,40,37);(1,2,3,4)(5,6,8,7)(9,12,15,13)(10,17,19,25)(11,21,24,22)(14,26,29,27)(16,30,31,32)(18,33,35,40)(20,36,34,37)(23,38,28,39);(5,7,8,6)(9,13,15,12)(10,17,19,25)(11,22,24,21)(14,27,29,26)(16,30,31,32)(18,33,35,40)(20,36,34,37)(23,39,28,38);(12,13)(17,25)(21,22)(26,27)(30,32)(33,40)(36,37)(38,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6372779);