#420383

order	512
nilpotency class	3
isoclinism	isoclinic class of #420383
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

1286 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)(3,14,18,40)(4,10)(5,25,28,11)(6,12)(7,30)(8,13)(9,17,33,19)(15,41,44,34)(16,45)(20,48,43,31)(21,35,51,54)(22,36)(23,53)(24,37)(26,32)(27,55,29,39)(38,47)(42,62)(46,58)(49,60)(50,61,52,64)(56,57)(59,63);(1,3)(2,9)(4,20)(5,26)(6,19)(7,17)(8,18)(10,34)(11,38)(12,40)(13,33)(14,30)(15,23)(16,27)(21,42)(22,44)(24,43)(25,45)(28,46)(29,47)(31,36)(32,39)(35,60)(37,41)(48,53)(49,50)(51,63)(52,56)(54,57)(55,58)(59,61)(62,64);(1,4,6,22,8,24,7,23)(2,10,12,36,13,37,30,53)(3,15,17,43,18,44,19,20)(5,21,27,50,28,51,29,52)(9,31,14,34,33,48,40,41)(11,35,39,61,25,54,55,64)(16,42,26,56,47,63,46,49)(32,57,38,59,58,60,45,62);(1,5,6,27,8,28,7,29)(2,11,12,39,13,25,30,55)(3,16,17,26,18,47,19,46)(4,21,22,50,24,51,23,52)(9,32,14,38,33,58,40,45)(10,35,36,61,37,54,53,64)(15,42,43,56,44,63,20,49)(31,57,34,59,48,60,41,62)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=e,e^2=i,g^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,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319923, #72319954, #72320069
maximal normal subgroup of	#72319923, #72319954, #72320069

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327579);