#99990

order	512
nilpotency class	3
isoclinism	isoclinic class of #99979
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	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

468 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cgi,d^b=dh,f^b=fh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7186);