#146417

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	#1869
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

880 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)(7,21,25,50,23,40,56,47,26,38,34,60,24,14,36,39)(8,27,30,58,29,44,59,48,31,42,35,61,28,15,41,43)(22,51,53,64,52,65,62,68,55,37,54,63,57,49,66,67);(2,4)(5,7,11,25,12,23,32,56,13,26,10,34,9,24,33,36)(6,14,18,39,19,21,45,50,20,40,17,47,16,38,46,60)(8,22,30,54,29,52,59,66,31,55,35,53,28,57,41,62)(15,37,43,64,27,49,58,68,44,51,48,63,42,65,61,67);(5,8)(6,15)(7,22)(9,28)(10,35)(11,30)(12,29)(13,31)(14,37)(16,42)(17,48)(18,43)(19,27)(20,44)(21,49)(23,52)(24,57)(25,54)(26,55)(32,59)(33,41)(34,53)(36,62)(38,65)(39,64)(40,51)(45,58)(46,61)(47,63)(50,68)(56,66)(60,67)

Polycyclic presentation:

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

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, 53613);