#295115

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

796 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)(11,15)(12,21)(13,17)(14,18)(16,22)(19,24)(20,23)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,39)(38,40);(2,5)(6,7)(9,11)(10,15)(12,22)(13,19)(14,20)(16,21)(17,24)(18,23)(26,30)(27,34)(29,36)(31,38);(9,12,13,11,14,19,22,20)(10,15,16,21,18,23,17,24)(25,27,29,35,28,34,36,40)(26,31,32,33,30,38,37,39);(1,3,4,8)(2,6,5,7)(26,30)(31,38)(32,37)(33,39)

Polycyclic presentation:

2^9|c^2=f,d^2=ei,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, 202311);