#295128

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202324);