#500735

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1017 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)(5,6,14,22)(7,23,28,55)(8,29,34,49)(9,20,35,19)(10,21,13,18)(11,36,12,17)(15,41,46,26)(16,33,50,31)(24,58,63,67)(25,45,27,44)(30,66,61,57)(32,54,42,52)(37,51,39,53)(38,56,40,48)(43,59,65,64)(47,60,68,62);(5,7,10,26,14,28,13,41)(6,15,18,23,22,46,21,55)(8,30,32,59,34,61,42,64)(9,25,12,39,35,27,11,37)(16,47,48,67,50,68,56,58)(17,44,20,53,36,45,19,51)(24,40,60,31,63,38,62,33)(29,65,52,66,49,43,54,57);(2,4)(5,8)(6,16)(7,24)(9,33)(10,32)(11,38)(12,40)(13,42)(14,34)(15,43)(17,49)(18,48)(19,52)(20,54)(21,56)(22,50)(23,57)(25,64)(26,60)(27,59)(28,63)(29,36)(30,39)(31,35)(37,61)(41,62)(44,58)(45,67)(46,65)(47,53)(51,68)(55,66);(5,9,13,11,14,35,10,12)(6,17,21,19,22,36,18,20)(7,25,41,37,28,27,26,39)(8,31,42,40,34,33,32,38)(15,44,55,51,46,45,23,53)(16,29,56,54,50,49,48,52)(24,59,62,30,63,64,60,61)(43,67,66,47,65,58,57,68)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407931);