#148890

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#1808
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 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)(8,25,28,56)(9,30)(10,15)(11,34)(12,18)(13,21)(14,27,40,60)(16,32)(17,31)(20,47,50,35)(22,44)(23,36)(24,46)(26,57,59,41)(29,61,58,39)(33,42)(37,54)(38,53)(43,49,64,68)(45,52)(48,55,67,65)(51,62,66,63);(1,2)(3,4)(5,7,10,22,11,23,32,53,12,24,33,54,31,52,9,21)(6,13,30,45,17,37,42,46,18,38,16,36,34,44,15,19)(8,20,27,49,29,51,59,67,28,50,60,68,58,66,26,48)(14,35,56,65,41,62,61,64,40,47,25,55,57,63,39,43);(2,4)(5,8,9,26,31,58,33,60,12,28,32,59,11,29,10,27)(6,14,15,39,34,57,16,25,18,40,42,61,17,41,30,56)(7,20,21,48,52,66,54,68,24,50,53,67,23,51,22,49)(13,35,19,43,44,63,36,55,38,47,46,64,37,62,45,65)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,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, 56086);