#6467848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
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
prime radical	#60704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(11,13)(12,17)(14,19)(15,20)(16,22)(18,26)(21,27)(23,30)(24,31)(25,38)(28,33)(29,35)(32,37)(34,39)(36,40);(9,11,14,22,15,24,26,23)(10,16,18,13,20,30,19,31)(12,25,29,21,28,37,39,36)(17,27,34,38,33,40,35,32);(2,5)(6,8)(9,12,26,39,15,28,14,29)(10,17,18,34,20,33,19,35)(11,21,22,37,24,36,23,25)(13,27,16,32,31,40,30,38);(9,13,14,30,15,31,26,16)(10,11,19,23,20,24,18,22)(12,27,29,38,28,40,39,32)(17,21,35,25,33,36,34,37);(1,3,4,7)(2,6,5,8)(12,28)(17,33)(21,36)(25,37)(27,40)(29,39)(32,38)(34,35)

Polycyclic presentation:

2^9|b^2=f,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,h^b=hi,e^c=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, 6375044);