#148860

order	512
nilpotency class	4
isoclinism	isoclinic class of #129558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

488 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:

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

Polycyclic presentation:

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

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, 56056);