#148344

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	#2033
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

650 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=i,g^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, 55540);