#493141

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=hi,f^2=h,h^2=i,b^a=be,e^b=eh,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400337);