#498374

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
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

601 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)(5,6,11,19,10,37,36,52,13,20,9,17,12,18,29,48)(7,21,25,43,38,46,35,14,26,42,23,41,24,15,39,44)(8,27,32,49,31,50,63,67,34,16,30,51,33,58,62,68)(22,53,55,65,60,45,61,66,56,47,28,59,54,57,64,40);(1,3,4,2)(5,7)(6,14)(8,28)(9,35)(10,24)(11,39)(12,38)(13,26)(15,17)(16,47)(18,41)(19,46)(20,44)(21,52)(22,30)(23,36)(25,29)(27,53)(31,61)(32,56)(33,64)(34,55)(37,43)(40,49)(42,48)(45,58)(50,57)(51,66)(54,62)(59,67)(60,63)(65,68);(5,7)(6,15)(8,22)(9,23)(10,38)(11,25)(12,24)(13,26)(14,17)(16,45)(18,42)(19,44)(20,46)(21,37)(27,57)(28,30)(29,39)(31,60)(32,55)(33,54)(34,56)(35,36)(40,49)(41,48)(43,52)(47,58)(50,53)(51,66)(59,68)(61,63)(62,64)(65,67);(5,8,29,62,12,33,9,30,13,34,36,63,10,31,11,32)(6,16,48,67,18,50,17,49,20,27,52,68,37,58,19,51)(7,22,39,64,24,54,23,28,26,56,35,61,38,60,25,55)(14,40,46,57,43,59,21,47,44,66,15,45,41,65,42,53)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405570);