#241097

order	512
nilpotency class	3
isoclinism	isoclinic class of #241096
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148293);