#494227

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1850
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341733
maximal normal subgroup of	#72341733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401423);