#355146

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13776
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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)(3,13,16,8)(4,18)(5,22,23,10)(6,12)(7,11)(9,20)(14,29,37,26)(15,27)(17,35,41,28)(19,30,44,42)(21,43)(24,47,48,32)(25,33)(31,46)(34,39)(36,49)(38,53,58,50)(40,51)(45,54,61,60)(52,57)(55,59)(56,62)(63,64);(1,3,6,17,7,16,25,41)(2,8,12,28,11,13,33,35)(4,14,21,38,20,37,46,58)(5,15,24,40,23,39,48,59)(9,29,31,53,18,26,43,50)(10,27,32,51,22,34,47,55)(19,36,45,56,44,57,61,64)(30,52,54,63,42,49,60,62);(1,4)(2,9,11,18)(3,14)(5,19)(6,21)(7,20)(8,26,13,29)(10,30,22,42)(12,31,33,43)(15,36)(16,37)(17,38)(23,44)(24,45)(25,46)(27,49,34,52)(28,50,35,53)(32,54,47,60)(39,57)(40,56)(41,58)(48,61)(51,62,55,63)(59,64);(1,5)(2,10)(3,15)(4,19)(6,24)(7,23)(8,27)(9,30)(11,22)(12,32)(13,34)(14,36)(16,39)(17,40)(18,42)(20,44)(21,45)(25,48)(26,49)(28,51)(29,52)(31,54)(33,47)(35,55)(37,57)(38,56)(41,59)(43,60)(46,61)(50,62)(53,63)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317348, #72317365
maximal normal subgroup of	#72317348, #72317365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262342);