#498370

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 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)(5,6,11,19,10,36,28,47,13,20,9,17,12,18,35,52)(7,21,25,42,37,45,34,14,26,41,23,40,24,15,38,43)(8,16,31,50,30,61,59,67,33,51,29,48,32,49,60,68)(22,46,54,64,57,66,58,39,55,63,27,56,53,44,62,65);(1,3,4,2)(5,7)(6,14)(8,27)(9,34)(10,24)(11,38)(12,37)(13,26)(15,17)(16,46)(18,40)(19,45)(20,43)(21,47)(22,29)(23,28)(25,35)(30,58)(31,55)(32,62)(33,54)(36,42)(39,48)(41,52)(44,61)(49,66)(50,65)(51,63)(53,60)(56,67)(57,59)(64,68);(5,7)(6,15)(8,22)(9,23)(10,37)(11,25)(12,24)(13,26)(14,17)(16,44)(18,41)(19,43)(20,45)(21,36)(27,29)(28,34)(30,57)(31,54)(32,53)(33,55)(35,38)(39,48)(40,52)(42,47)(46,61)(49,63)(50,65)(51,66)(56,68)(58,59)(60,62)(64,67);(5,8,28,59,12,32,11,31,13,33,35,60,10,30,9,29)(6,16,47,67,18,49,19,50,20,51,52,68,36,61,17,48)(7,22,34,58,24,53,25,54,26,55,38,62,37,57,23,27)(14,39,15,44,42,64,41,63,43,65,45,66,40,56,21,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,d^2=eh,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,f^c=fi

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, 405566);