#310031

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

591 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)(3,6)(4,5)(7,8)(9,10,13,18,14,19,23,17)(11,20,21,24,12,16,22,15)(25,26)(27,30)(28,31)(29,32)(33,40)(34,38)(35,37)(36,39);(9,11,14,12)(10,15,19,24)(13,22,23,21)(16,17,20,18)(25,27,28,34)(26,30,31,38)(29,33,36,35)(32,40,39,37);(2,5)(6,8)(9,12,14,11)(10,16,19,20)(13,22,23,21)(15,17,24,18)(26,31)(27,33)(29,36)(30,37)(34,35)(38,40);(1,3,4,7)(2,6,5,8)(10,17)(13,23)(15,20)(16,24)(18,19)(21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217227);