#331446

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 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)(9,10,15,21,14,12,13,17)(11,16,22,24,18,20,19,23)(25,26,30,35,28,33,29,34)(27,36,39,32,37,31,38,40);(2,5)(6,8)(9,11)(10,16)(12,20)(13,19)(14,18)(15,22)(17,23)(21,24)(25,27,29,38,28,37,30,39)(26,31,34,32,33,36,35,40);(3,7)(6,8)(9,12,14,10)(11,16,18,20)(13,17,15,21)(19,24,22,23)(25,27,28,37)(26,32,33,40)(29,39,30,38)(31,35,36,34);(1,3,4,7)(2,6,5,8)(11,18)(16,20)(19,22)(23,24)(27,37)(31,36)(32,40)(38,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,c^2=h,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238642);