#419340

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326536);