#427875

order	512
nilpotency class	3
isoclinism	isoclinic class of #427874
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15375
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335071);