#6432886

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432881
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340082);