#129126

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₄×#106
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

232 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36322);