#148343

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ 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)(7,19)(8,26,30,14)(9,31)(10,18)(11,17)(12,16)(13,22)(15,20)(21,53,55,47)(23,50)(24,49)(25,48)(27,61,59,68)(28,64,32,65)(29,42,35,46)(33,44)(34,45)(36,56,67,54)(37,58)(38,51)(39,57)(40,63,60,66)(41,62,43,52);(1,2)(3,4)(5,7,20,51,11,24,34,58,12,25,33,57,10,23,9,22)(6,13,31,50,18,39,44,48,16,37,45,49,17,38,15,19)(8,27,62,56,29,55,32,66,30,59,52,54,35,21,28,63)(14,40,65,47,42,67,43,61,26,60,64,53,46,36,41,68);(2,4)(5,8)(6,14)(7,21)(9,32)(10,35)(11,29)(12,30)(13,36)(15,43)(16,26)(17,46)(18,42)(19,47)(20,52)(22,56)(23,59)(24,27)(25,55)(28,34)(31,64)(33,62)(37,67)(38,60)(39,40)(41,44)(45,65)(48,53)(49,68)(50,61)(51,66)(54,58)(57,63)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^c=di,f^c=fi,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, 55539);