#493778

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1056 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)(5,6,10,15,11,17,31,16)(7,12,21,35,19,33,48,36)(8,22,24,40,25,54,30,53)(9,26,28,14,29,41,57,42)(13,27,37,23,39,55,38,56)(18,43,46,62,44,64,52,65)(20,49,51,34,47,63,68,61)(32,50,59,45,58,67,60,66);(1,2,3,4)(5,7)(6,12)(8,18)(9,20)(10,21)(11,19)(13,32)(14,34)(15,35)(16,36)(17,33)(22,43)(23,45)(24,46)(25,44)(26,49)(27,50)(28,51)(29,47)(30,52)(31,48)(37,59)(38,60)(39,58)(40,62)(41,63)(42,61)(53,65)(54,64)(55,67)(56,66)(57,68);(5,8,11,25)(6,13,17,39)(7,18,19,44)(9,27,29,55)(10,30,31,24)(12,32,33,58)(14,40,42,53)(15,38,16,37)(20,50,47,67)(21,52,48,46)(22,26,54,41)(23,28,56,57)(34,62,61,65)(35,60,36,59)(43,49,64,63)(45,51,66,68);(7,19)(12,33)(18,44)(20,47)(21,48)(32,58)(34,61)(35,36)(43,64)(45,66)(46,52)(49,63)(50,67)(51,68)(59,60)(62,65)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400974);