#296548

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=f,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,g^c=ghi

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, 203744);