#6467251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,19)(18,21)(20,26)(22,29)(23,28)(24,27)(25,31)(30,34)(32,35)(33,37)(36,38)(39,40);(10,15)(11,20)(12,24)(13,21)(16,28)(17,31)(18,19)(22,36)(23,34)(25,35)(29,39)(37,38);(2,5)(6,8)(9,11,21,34,14,23,13,20)(10,16,18,30,19,28,15,26)(12,22,35,40,25,36,24,33)(17,29,32,39,31,38,27,37);(9,12,13,24,14,25,21,35)(10,17,15,27,19,31,18,32)(11,22,20,33,23,36,34,40)(16,29,26,37,28,38,30,39);(1,3,4,7)(2,6,5,8)(12,25)(17,31)(22,36)(24,35)(27,32)(29,38)(33,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374447);