#351124

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

587 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)(9,10,14,16)(11,17,18,24)(12,13,20,21)(15,22,23,19)(25,26)(27,30)(28,31)(29,33)(32,36)(34,37)(35,38)(39,40);(2,5)(6,8)(9,11,13,19,14,18,21,22)(10,15,12,17,16,23,20,24)(25,27)(26,30)(28,35)(29,34)(31,38)(32,39)(33,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12,13,16,14,20,21,10)(11,17,19,23,18,24,22,15)(27,34)(30,37)(35,40)(38,39);(11,18)(15,23)(17,24)(19,22)(26,31)(28,36)(30,38)(32,33)(35,40)(37,39)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=dh,g^d=gi

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, 258320);