#147663

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
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

1008 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)(5,6,10,15,11,16,33,17)(7,18,21,41,22,46,32,47)(8,23,25,40,27,44,26,13)(9,28,30,14,31,42,62,43)(12,29,35,20,36,53,37,54)(19,48,50,65,52,66,51,45)(24,57,59,67,61,39,60,56)(34,63,55,68,38,49,64,58);(5,7,11,22)(6,12,16,36)(8,19,27,52)(9,29,31,53)(10,32,33,21)(13,38,40,34)(14,41,43,47)(15,37,17,35)(18,28,46,42)(20,30,54,62)(23,55,44,64)(24,58,61,68)(25,51,26,50)(39,48,57,66)(45,67,65,56)(49,60,63,59);(2,4)(5,8)(6,13)(7,19)(9,24)(10,26)(11,27)(12,34)(14,39)(15,44)(16,40)(17,23)(18,45)(20,49)(21,51)(22,52)(25,33)(28,56)(29,58)(30,60)(31,61)(32,50)(35,64)(36,38)(37,55)(41,66)(42,67)(43,57)(46,65)(47,48)(53,68)(54,63)(59,62)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=i,g^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=dhi,e^b=ei,g^b=gi,h^b=hi,e^c=ei,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, 54859);