#431461

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 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:

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

Polycyclic presentation:

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

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, 338657);