#420395

order	512
nilpotency class	3
isoclinism	isoclinic class of #420395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15277
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1284 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318867, #72318890, #72318981
maximal normal subgroup of	#72318867, #72318890, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327591);