#148283

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1807
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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)(7,17,20,38)(8,22,24,13)(9,26)(10,29)(11,15)(12,19,32,44)(14,27)(16,30)(18,40)(21,47,46,39)(23,51,52,55)(25,35,54,50)(28,56)(31,41)(33,60,58,45)(34,53,49,67)(36,57)(37,43)(42,61)(48,65)(59,68)(62,64)(63,66);(1,2)(3,4)(5,7,9,19,10,21,27,45,11,20,28,44,30,46,57,60)(6,12,14,17,16,33,26,39,15,32,36,38,29,58,56,47)(8,18,23,41,25,42,53,64,24,43,52,65,54,66,67,68)(13,31,34,37,35,59,55,61,22,48,49,40,50,62,51,63);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,31)(14,34)(15,22)(16,35)(17,37)(19,41)(20,43)(21,42)(26,55)(27,53)(28,52)(29,50)(30,54)(32,48)(33,59)(36,49)(38,40)(39,61)(44,65)(45,64)(46,66)(47,63)(51,56)(57,67)(58,62)(60,68)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,g^b=gi,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, 55479);