#499034

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1040 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406230);