#98129

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

450 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5325);