#99947

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

469 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,4,6,5,8,3,7)(9,10,13,15,12,11,14,16)(17,18)(19,22,26,31)(20,23)(21,24)(25,28)(27,29,32,30);(2,6)(3,4)(7,8)(9,11,13,16,12,10,14,15)(17,19)(18,22,24,31)(20,27)(21,26)(23,29,25,30)(28,32);(1,3)(4,5)(6,7)(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eh,f^c=fgh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7143);