#316154

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=ef,d^2=e,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

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, 223350);