#146389

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	#1779
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

1076 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^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, 53585);