#454981

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16258
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1427 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)(3,14,17,9)(4,10)(5,23,24,56)(6,27)(7,13)(8,12)(11,26,40,25)(15,30,44,41)(16,46)(18,32,28,33)(19,49,45,34)(20,51,52,55)(21,38)(22,36)(29,37)(31,47)(35,53,39,58)(42,57)(43,61)(48,54)(50,59)(60,63)(62,64);(1,3)(2,9)(4,19)(5,16)(6,28)(7,18)(8,17)(10,34)(11,31)(12,14)(13,33)(15,21)(20,50)(22,45)(23,54)(24,48)(25,57)(26,47)(27,32)(29,44)(30,37)(35,61)(36,49)(38,41)(39,62)(40,42)(43,53)(46,56)(51,60)(52,63)(55,59)(58,64);(1,4)(2,10)(3,15)(5,20)(6,29)(7,21)(8,22)(9,30)(11,39)(12,36)(13,38)(14,41)(16,43)(17,44)(18,45)(19,28)(23,55)(24,52)(25,58)(26,53)(27,37)(31,60)(32,49)(33,34)(35,40)(42,59)(46,61)(47,63)(48,64)(50,57)(51,56)(54,62);(1,5)(2,11,12,40)(3,16)(4,20)(6,25)(7,26)(8,24)(9,31,14,42)(10,35,36,39)(13,23,27,56)(15,43)(17,48)(18,47)(19,50)(21,53)(22,52)(28,57)(29,58)(30,59,41,60)(32,46,33,54)(34,61,49,62)(37,55,38,51)(44,64)(45,63)

Polycyclic presentation:

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

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, 362177);