#101783

order	512
nilpotency class	3
isoclinism	isoclinic class of #101687
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 416 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) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=gh,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8979);