#148877

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2065
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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)(7,18,21,36)(8,23)(9,28)(10,31)(11,34)(12,16)(13,20,37,32)(14,25)(15,30)(17,29)(19,46,48,63)(22,52,51,45)(24,40)(26,60)(27,54)(33,42)(35,49,39,64)(38,61,44,50)(41,57)(43,58,53,47)(55,56)(59,62)(65,67,66,68);(5,7,9,20,11,22,30,50,12,21,10,32,29,51,33,61)(6,13,15,18,17,38,28,45,16,37,42,36,34,44,31,52)(8,19,24,47,26,49,57,67,27,48,25,58,56,64,59,68)(14,39,23,53,41,46,55,66,40,35,54,43,62,63,60,65);(2,4)(5,8,9,24,11,26,30,57,12,27,10,25,29,56,33,59)(6,14,15,23,17,41,28,55,16,40,42,54,34,62,31,60)(7,19,20,47,22,49,50,67,21,48,32,58,51,64,61,68)(13,35,18,43,38,63,45,65,37,39,36,53,44,46,52,66)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=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, 56073);