#123304

order	512
nilpotency class	4
isoclinism	isoclinic class of #123275
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1619
commutator, Frattini, cosocle	C₈²	C₂³
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

227 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 227 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,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,d^b=df,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fi,g^c=gi

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, 30500);