#135702

order	512
nilpotency class	4
isoclinism	isoclinic class of #135685
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#487
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

189 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=dg,d^2=hi,e^2=gi,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,c^b=ci,f^b=fghi,f^c=fghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42898);