#234724

order	512
nilpotency class	3
isoclinism	isoclinic class of #232331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1285 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,10,14)(7,12,16,20)(8,13,9,11)(15,19,17,18)(21,22,26,30)(23,31,34,44)(24,28,35,42)(25,37,39,29)(27,38,41,33)(32,45,46,52)(36,49,48,43)(40,50,51,47);(6,11)(8,9)(12,19)(13,14)(15,17)(18,20)(21,23,26,34)(22,27,30,41)(24,32,35,46)(25,38,39,33)(28,40,42,51)(29,44,37,31)(36,50,48,47)(43,52,49,45);(2,4)(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,43)(30,42)(31,45)(33,47)(34,46)(37,49)(38,50)(39,48)(41,51)(44,52);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,19,20,18)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(45,52)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141920);