#135660

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42856);