#493520

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,17,11,32)(7,18,21,44,22,45,33,41)(8,14,25,40,23,38,26,55)(9,27,29,42,31,57,30,15)(13,28,35,49,37,58,36,20)(19,43,48,65,46,64,56,63)(24,51,53,62,50,67,54,39)(34,52,60,66,59,68,61,47);(5,7)(6,13)(8,19)(9,28)(10,21)(11,33)(12,22)(14,34)(15,41)(16,35)(17,37)(18,27)(20,30)(23,46)(24,52)(25,48)(26,56)(29,49)(31,58)(32,36)(38,59)(39,63)(40,60)(42,44)(43,51)(45,57)(47,54)(50,68)(53,66)(55,61)(62,65)(64,67);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,23)(13,34)(15,39)(16,40)(17,38)(18,43)(20,47)(21,48)(22,46)(27,51)(28,52)(29,53)(30,54)(31,50)(32,55)(33,56)(35,60)(36,61)(37,59)(41,63)(42,62)(44,65)(45,64)(49,66)(57,67)(58,68);(8,23)(14,38)(19,46)(24,50)(25,26)(34,59)(39,62)(40,55)(43,64)(47,66)(48,56)(51,67)(52,68)(53,54)(60,61)(63,65)

Polycyclic presentation:

2^9|a^2=g,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323138
maximal normal subgroup of	#72323138
maximal quotient of	#72323010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400716);