#493769

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1076 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400965);