#149215

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1852
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 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:

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

Polycyclic presentation:

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

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, 56411);