#297150

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^2=f,d^2=ehi,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,d^c=di,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204346);