#418771

order	512
nilpotency class	3
isoclinism	isoclinic class of #418770
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15266
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1874 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)(3,15,20,34)(4,21,25,42)(5,26,29,11)(6,12)(7,13)(8,14)(9,18,35,19)(10,24,38,31)(16,32)(17,43)(22,36)(23,51)(27,33)(28,41,30,40)(37,53)(39,48)(44,57,63,64)(45,58)(46,59)(47,60)(49,54)(50,61,56,62)(52,55);(1,3)(2,9)(4,22)(5,27)(6,19)(7,18)(8,20)(10,36)(11,39)(12,34)(13,15)(14,35)(16,24)(17,28)(21,32)(23,44)(25,45)(26,43)(29,47)(30,48)(31,46)(33,40)(37,57)(38,58)(41,60)(42,59)(49,62)(50,52)(51,61)(53,56)(54,63)(55,64);(1,4,8,25)(2,10,14,38)(3,16,20,46)(5,23,29,54)(6,24,7,31)(9,32,35,59)(11,37,26,55)(12,21,13,42)(15,36,34,58)(17,44,48,63)(18,22,19,45)(27,56,47,50)(28,52,30,53)(33,57,60,64)(39,62,43,61)(40,51,41,49);(1,5)(2,11)(3,17)(4,23)(6,30)(7,28)(8,29)(9,33)(10,37)(12,41)(13,40)(14,26)(15,43)(16,44)(18,47)(19,27)(20,48)(21,49)(22,50)(24,53)(25,54)(31,52)(32,57)(34,39)(35,60)(36,61)(38,55)(42,51)(45,56)(46,63)(58,62)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317848, #72318497, #72318589
maximal normal subgroup of	#72317848, #72318497, #72318589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325967);