#245925

order	512
nilpotency class	3
isoclinism	isoclinic class of #245903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1169 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153121);