#254840

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

822 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)(27,37)(28,32)(29,36)(30,35)(31,34)(33,39)(38,40);(2,5)(6,8)(9,11,15,20)(10,12,19,23)(13,16,14,24)(17,21,18,22)(25,27,30,28)(26,32,35,37)(29,38,31,39)(33,34,40,36);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,20)(19,24)(25,28,30,27)(26,33,35,40)(29,39,31,38)(32,36,37,34);(2,5)(6,8)(9,13,15,14)(10,17,19,18)(11,16,20,24)(12,21,23,22)(25,29,30,31)(26,34,35,36)(27,38,28,39)(32,40,37,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,d^2=h,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, 162036);