#509177

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33526	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

772 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,e^d=eh,f^d=fi,h^d=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, 416373);