#97560

order	512
nilpotency class	3
isoclinism	isoclinic class of #97550
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,e^b=eh,f^b=fg,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4756);