#380131

order	512
nilpotency class	3
isoclinism	isoclinic class of #380130
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14259
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287327);