#495833

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gi,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,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, 403029);