#259986

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 167182);