#498574

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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)(5,6,10,16,35,17,29,41,12,19,28,14,11,18,8,26)(7,20,21,48,23,36,50,39,25,49,51,38,24,47,34,13)(9,15,31,43,62,44,57,66,33,46,56,40,32,45,27,55)(22,42,30,58,52,63,59,65,54,68,61,64,53,67,60,37);(1,3,4,2)(5,7)(6,13)(8,21)(9,30)(10,34)(11,23)(12,25)(14,36)(15,42)(16,47)(17,38)(18,48)(19,39)(20,26)(22,31)(24,35)(27,52)(28,50)(29,51)(32,59)(33,61)(37,43)(40,65)(41,49)(44,67)(45,63)(46,68)(53,57)(54,56)(55,58)(60,62)(64,66);(5,8,11,28,12,29,35,10)(6,14,17,26,19,16,18,41)(7,21,23,50,25,51,24,34)(9,27,32,56,33,57,62,31)(13,36,38,20,39,47,48,49)(15,40,44,55,46,43,45,66)(22,30,52,59,54,61,53,60)(37,63,64,42,65,67,58,68);(5,9,29,57,11,32,10,31,12,33,8,27,35,62,28,56)(6,15,41,66,18,45,16,43,19,46,26,55,17,44,14,40)(7,22,50,59,24,53,21,30,25,54,34,60,23,52,51,61)(13,37,36,63,38,64,20,42,39,65,47,67,48,58,49,68)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=ehi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405770);