#405388

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14912
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1232 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)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,23)(15,30)(16,41)(18,33)(19,32)(21,34)(22,52)(24,36)(25,50)(26,38)(27,37)(28,39)(31,45)(35,54)(40,43)(42,62)(44,48)(46,57)(47,58)(49,55)(51,61)(53,60)(56,63)(59,64);(1,3)(2,9)(4,21,25,43)(5,16)(6,18)(7,17)(8,19)(10,34,36,40)(11,31)(12,32)(13,33)(14,29)(15,24,44,23)(20,48,50,30)(22,51,55,64)(26,47)(27,45)(28,46)(35,59,60,61)(37,57)(38,41)(39,58)(42,54,63,53)(49,62,52,56);(1,4,6,23,8,25,7,24)(2,10,12,20,13,36,29,50)(3,15,17,21,19,44,18,43)(5,22,26,53,28,55,27,54)(9,30,14,40,33,48,32,34)(11,35,37,49,39,60,38,52)(16,42,45,51,46,63,47,64)(31,56,41,61,58,62,57,59);(1,5)(2,11)(3,16)(4,22)(6,26)(7,27)(8,28)(9,31)(10,35)(12,37)(13,39)(14,41)(15,42)(17,45)(18,47)(19,46)(20,49)(21,51)(23,53)(24,54)(25,55)(29,38)(30,56)(32,57)(33,58)(34,59)(36,60)(40,61)(43,64)(44,63)(48,62)(50,52)

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=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, 312584);