#148327

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
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:

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

Polycyclic presentation:

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

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, 55523);