#395953

order	512
nilpotency class	3
isoclinism	isoclinic class of #395952
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

1287 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,e^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,f^c=fi,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, 303149);