#123391

order	512
nilpotency class	4
isoclinism	isoclinic class of #123383
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	C₂×D₈	D₈
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

153 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 153 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=df,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gi,c^b=ch,e^c=eg,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30587);