#495094

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi,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, 402290);