#395702

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
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

1815 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)(3,15)(4,21)(5,27,29,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,42)(17,46,48,41)(19,34)(20,33)(22,31)(23,50,53,59)(24,37)(26,36)(28,38,30,39)(32,47,55,49)(35,52,56,54)(40,45)(43,61,51,57)(44,58)(60,62,64,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,45,26,16,25,44)(5,17,28,47,29,48,30,49)(10,31,36,58,37,42,21,40)(11,32,38,41,27,55,39,46)(23,51,52,63,53,43,54,62)(35,57,59,64,56,61,50,60);(1,4)(2,10)(3,16)(5,23)(6,25)(7,24)(8,26)(9,31)(11,35)(12,21)(13,36)(14,37)(15,40)(17,43)(18,45)(19,44)(20,22)(27,56)(28,54)(29,53)(30,52)(32,57)(33,58)(34,42)(38,50)(39,59)(41,60)(46,64)(47,63)(48,51)(49,62)(55,61);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,35)(12,38)(13,39)(14,27)(15,41)(16,43)(18,47)(19,49)(20,48)(21,50)(22,51)(24,52)(25,54)(26,53)(31,57)(33,46)(34,55)(36,59)(37,56)(40,60)(42,61)(44,62)(45,63)(58,64)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=i,f^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,f^c=fi,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, 302898);