#147677

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1000 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)(7,18)(8,24,28,14)(9,29)(10,17)(11,35)(12,16)(13,20)(15,32)(19,49,51,47)(21,43)(22,55)(23,48)(25,58,54,67)(26,60,63,64)(27,42,34,41)(30,45)(31,44)(33,46)(36,50,66,61)(37,53)(38,52)(39,59,56,68)(40,62,57,65);(5,7,12,23)(6,13,16,37)(8,25,28,54)(9,30,33,52)(10,22,11,21)(14,39,24,56)(15,43,44,55)(17,45,35,38)(18,46,48,29)(19,34,51,27)(20,31,53,32)(26,61,63,50)(36,42,66,41)(40,47,57,49)(58,64,67,60)(59,65,68,62);(2,4)(5,8)(6,14)(7,19)(9,26)(10,34)(11,27)(12,28)(13,36)(15,40)(16,24)(17,42)(18,47)(20,50)(21,54)(22,25)(23,51)(29,64)(30,59)(31,65)(32,62)(33,63)(35,41)(37,66)(38,56)(39,45)(43,58)(44,57)(46,60)(48,49)(52,68)(53,61)(55,67)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=dh,f^b=fi,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54873);