#457761

order	512
nilpotency class	3
isoclinism	isoclinic class of #457760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16301
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364957);