#135631

order	512
nilpotency class	4
isoclinism	isoclinic class of #135606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1824
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#103
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

237 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,e^2=gh,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,d^c=di,e^c=eh,f^c=fgi,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42827);