#234726

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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1233 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,17)(18,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,39)(32,40)(33,41)(34,43)(35,44)(36,45)(37,46)(38,47)(42,49)(48,51)(50,52);(5,7)(6,12)(8,11)(9,17)(10,18)(13,14)(15,20)(16,19)(21,23,26,33)(22,27,30,41)(24,31,35,49)(25,36,38,32)(28,42,44,39)(29,45,47,40)(34,50,37,48)(43,51,46,52);(2,4)(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14)(21,24)(22,28)(23,31)(25,37)(26,35)(27,39)(29,46)(30,44)(32,50)(33,49)(34,38)(36,48)(40,52)(41,42)(43,47)(45,51);(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,25,26,38)(22,29,30,47)(23,32,33,36)(24,34,35,37)(27,40,41,45)(28,43,44,46)(31,48,49,50)(39,51,42,52)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,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, 141922);