#360230

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13872
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267426);