#493215

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

748 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=i,f^2=h,h^2=i,b^a=be,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400411);