#476115

order	512
nilpotency class	3
isoclinism	isoclinic class of #476112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,24,37)(8,14,29,43)(9,15,33,47)(10,17,11,16)(13,23,38,22)(20,39,52,62)(21,49,54,44)(25,35,51,63)(26,56,58,40)(27,42,28,41)(30,60,55,36)(31,45,59,66)(32,48,34,46)(50,67,57,64)(53,65,68,61);(1,2)(3,4)(5,7,12,24)(6,13,18,38)(8,25,29,51)(9,30,33,55)(10,23,11,22)(14,39,43,62)(15,44,47,49)(16,37,17,19)(20,27,52,28)(21,34,54,32)(26,57,58,50)(31,53,59,68)(35,41,63,42)(36,48,60,46)(40,65,56,61)(45,64,66,67);(2,4)(5,8)(6,14)(7,20)(9,31)(10,28)(11,27)(12,29)(13,35)(15,45)(16,42)(17,41)(18,43)(19,39)(21,53)(22,25)(23,51)(24,52)(26,32)(30,50)(33,59)(34,58)(36,64)(37,62)(38,63)(40,46)(44,61)(47,66)(48,56)(49,65)(54,68)(55,57)(60,67);(2,4)(5,9)(6,15)(7,21)(8,26)(10,34)(11,32)(12,33)(13,36)(14,40)(16,48)(17,46)(18,47)(19,49)(20,50)(22,55)(23,30)(24,54)(25,53)(27,59)(28,31)(29,58)(35,61)(37,44)(38,60)(39,64)(41,66)(42,45)(43,56)(51,68)(52,57)(62,67)(63,65)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383311);