#147699

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
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

996 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)(7,19)(8,25,29,42)(9,30)(10,16)(11,17)(12,18)(13,21)(14,28,43,35)(15,33)(20,50,52,66)(22,44)(23,48)(24,49)(26,58,55,47)(27,41,62,67)(31,38)(32,46)(34,45)(36,59,56,68)(37,53)(39,54)(40,51,65,60)(57,64,63,61);(1,2)(3,4)(5,7,12,24)(6,13,18,39)(8,26,29,55)(9,31,34,53)(10,23,11,22)(14,40,43,65)(15,44,46,48)(16,38,17,37)(19,45,49,30)(20,35,52,28)(21,32,54,33)(25,56,42,36)(27,60,62,51)(41,58,67,47)(50,63,66,57)(59,64,68,61);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,28)(12,29)(13,36)(15,41)(16,42)(17,25)(18,43)(19,47)(21,51)(22,55)(23,26)(24,52)(30,63)(31,59)(32,64)(33,61)(34,62)(37,40)(38,65)(39,56)(44,66)(45,57)(46,67)(48,50)(49,58)(53,68)(54,60)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 54895);