#122651

order	512
nilpotency class	4
isoclinism	isoclinic class of #122564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

423 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=chi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29847);