#492946

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×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	C₂²×C₂²⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1268 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)(7,18)(8,14)(9,28)(10,30)(11,15)(12,17)(13,20)(16,32)(19,40)(21,48)(22,35)(23,42)(24,36)(25,52)(26,54)(27,37)(29,39)(31,56)(33,44)(34,49)(38,51)(41,50)(43,61)(45,65)(46,58)(47,57)(53,60)(55,67)(59,66)(62,63)(64,68);(5,7,10,21,11,22,31,49,12,23,32,50,9,20,29,47)(6,13,16,34,15,18,39,41,17,35,30,57,28,42,56,48)(8,19,26,45,27,46,55,66,24,43,51,63,25,44,53,64)(14,33,38,59,37,40,60,62,36,58,54,68,52,61,67,65);(1,3,4,2)(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,24)(13,33)(15,37)(16,38)(17,36)(18,40)(20,44)(21,45)(22,46)(23,43)(28,52)(29,53)(30,54)(31,55)(32,51)(34,59)(35,58)(39,60)(41,62)(42,61)(47,64)(48,65)(49,66)(50,63)(56,67)(57,68);(8,24)(14,36)(19,43)(25,27)(26,51)(33,58)(37,52)(38,54)(40,61)(44,46)(45,63)(53,55)(59,68)(60,67)(62,65)(64,66)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,e^a=ei,g^a=gi,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, 400142);