#254858

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

845 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)(9,10,15,19)(11,12,20,23)(13,17,14,18)(16,21,24,22)(25,26,29,33)(27,30,35,40)(28,31,37,32)(34,38,36,39);(2,5)(6,8)(9,11,15,20)(10,12,19,23)(13,16,14,24)(17,21,18,22)(25,27,29,35)(26,30,33,40)(28,36,37,34)(31,39,32,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(26,31)(28,37)(30,38)(32,33)(34,36)(39,40);(2,5)(6,8)(9,13,15,14)(10,17,19,18)(11,16,20,24)(12,21,23,22)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162054);