#430922

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15429
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338118);