#149003

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#1835
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=h,g^2=i,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, 56199);