#509211

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	#24290	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=fh,h^c=hi,e^d=ei,f^d=fh,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, 416407);