#395951

order	512
nilpotency class	3
isoclinism	isoclinic class of #395950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14729
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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)(4,19)(5,24,27,11)(6,28)(7,12)(8,13)(9,17)(10,22)(15,42)(16,46,49,41)(18,33)(20,51)(21,50,55,60)(23,37)(25,39,26,40)(29,34)(30,38)(31,45)(32,47,56,48)(35,44)(36,54,57,53)(43,61,63,58)(52,64,62,59);(1,3,6,17,8,18,7,29)(2,9,12,14,13,34,28,33)(4,20,22,15,23,44,30,45)(5,16,25,47,27,49,26,48)(10,35,19,42,38,51,37,31)(11,32,39,41,24,56,40,46)(21,52,53,43,55,62,54,63)(36,59,50,61,57,64,60,58);(1,4,6,22,8,23,7,30)(2,10,12,19,13,38,28,37)(3,15,17,44,18,45,29,20)(5,21,25,53,27,55,26,54)(9,31,14,35,34,42,33,51)(11,36,39,50,24,57,40,60)(16,43,47,62,49,63,48,52)(32,58,41,59,56,61,46,64);(1,5)(2,11)(3,16)(4,21)(6,25)(7,26)(8,27)(9,32)(10,36)(12,39)(13,24)(14,41)(15,43)(17,47)(18,49)(19,50)(20,52)(22,53)(23,55)(28,40)(29,48)(30,54)(31,58)(33,46)(34,56)(35,59)(37,60)(38,57)(42,61)(44,62)(45,63)(51,64)

Polycyclic presentation:

2^9|a^2=efi,b^2=e,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,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303147);