#494422

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1290 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)(3,4)(5,6,10,16,11,15,30,33,12,17,31,40,9,29,19,44)(7,18,22,42,23,36,50,59,24,43,51,63,21,13,32,35)(8,14,27,39,28,38,54,56,25,37,52,60,26,53,45,64)(20,41,48,62,49,57,66,67,46,61,65,68,47,34,55,58);(5,7,19,32,9,21,31,51,12,24,30,50,11,23,10,22)(6,13,33,42,29,43,16,35,17,36,44,63,15,18,40,59)(8,20,45,55,26,47,52,65,25,46,54,66,28,49,27,48)(14,34,56,62,53,61,39,58,37,57,64,68,38,41,60,67);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,25)(13,34)(15,38)(16,39)(17,37)(18,41)(19,45)(21,47)(22,48)(23,49)(24,46)(29,53)(30,54)(31,52)(32,55)(33,56)(35,58)(36,57)(40,60)(42,62)(43,61)(44,64)(50,66)(51,65)(59,67)(63,68);(8,25)(14,37)(20,46)(26,28)(27,52)(34,57)(38,53)(39,60)(41,61)(45,54)(47,49)(48,65)(55,66)(56,64)(58,67)(62,68)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,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, 401618);