#123418

order	512
nilpotency class	4
isoclinism	isoclinic class of #123412
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1712
commutator, Frattini, cosocle	C₈²	C₂³
prime radical	#1712	C₂²
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

225 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=dfi,c^2=egh,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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225191, #72225196, #72225200
maximal normal subgroup of	#72225191, #72225196, #72225200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30614);