#147665

order	512
nilpotency class	4
isoclinism	isoclinic class of #129565
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dh,e^b=ei,h^b=hi,d^c=di,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, 54861);