#146403

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^b=di,e^b=ei,f^b=fi,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, 53599);