#232218

order	512
nilpotency class	3
isoclinism	isoclinic class of #232215
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,32)(18,23)(19,22)(20,26)(25,41)(27,35)(28,40)(29,34)(30,39)(31,37)(33,38)(36,42)(43,44);(9,12)(13,15,19,28)(14,20,22,35)(16,29,31,43)(17,27,18,26)(21,36,37,41)(23,40,32,24)(25,33,42,30)(34,38,44,39);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16,19,31)(14,21,22,37)(15,25,28,42)(17,30,18,33)(20,34,35,44)(23,38,32,39)(24,41,40,36)(26,29,27,43);(14,22)(20,35)(21,37)(23,32)(24,40)(34,44)(36,41)(38,39)

Polycyclic presentation:

2^9|b^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,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, 139414);