#147667

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1786
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1024 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)(7,18,22,47)(8,23,26,45)(9,28)(10,15)(11,17)(12,20,37,57)(13,25,40,27)(14,31)(16,34)(19,49)(21,41,33,48)(24,42,30,39)(29,36,56,38)(32,43)(35,64)(44,54)(46,52)(50,66)(51,65)(53,68)(55,58)(59,61,63,62)(60,67);(1,2)(3,4)(5,7)(6,12)(8,19)(9,29)(10,33)(11,22)(13,35)(14,41)(15,38)(16,36)(17,37)(18,43)(20,54)(21,34)(23,58)(24,60)(25,52)(26,53)(27,51)(28,47)(30,64)(31,57)(32,56)(39,49)(40,67)(42,68)(44,48)(45,66)(46,59)(50,61)(55,62)(63,65);(2,4)(5,8)(6,13)(7,19)(9,30)(10,27)(11,26)(12,35)(14,42)(15,45)(16,23)(17,40)(18,46)(20,55)(21,52)(22,53)(24,32)(25,34)(28,63)(29,64)(31,61)(33,51)(36,58)(37,67)(38,66)(39,44)(41,68)(43,59)(47,65)(48,49)(50,57)(54,62)(56,60)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54863);