#259994

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

866 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)(9,10,15,11)(12,20,24,21)(13,19,14,17)(16,23,18,22)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(2,5)(6,7)(9,11,15,10)(12,21,24,20)(13,19,14,17)(16,23,18,22)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(9,12,15,24)(10,16,11,18)(13,23,14,22)(17,21,19,20)(25,28)(26,31)(27,33)(29,35)(30,36)(32,39)(34,40)(37,38);(1,3,4,8)(2,6,5,7)(28,35)(31,39)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=e,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=di,f^c=fh,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, 167190);