#427247

order	512
nilpotency class	3
isoclinism	isoclinic class of #427246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15368
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264914, #72264921, #72265017
maximal normal subgroup of	#72264914, #72264921, #72265017
maximal quotient of	#72265560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334443);