#395701

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14726
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1809 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302897);