#135793

order	512
nilpotency class	4
isoclinism	isoclinic class of #135758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#498
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

217 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225156, #72225276
maximal normal subgroup of	#72225156, #72225276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42989);