#135773

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#494
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,23,30,25,32,24,31)(19,33,37,48,39,27,38,44)(20,40,42,46,35,26,36,45)(21,29,22,28,41,43,34,47);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,34,42,23,37,21,35,25,39,22,36,24,38,41,20)(18,26,43,48,31,46,28,33,32,40,29,44,30,45,47,27);(2,7)(6,14)(8,12)(9,15)(17,20,41,38,24,36,22,39,25,35,21,37,23,42,34,19)(18,27,47,45,30,44,29,40,32,33,28,46,31,48,43,26)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=eg,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42969);