#144384

order	512
nilpotency class	4
isoclinism	isoclinic class of #144384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10336
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

873 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 32

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

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

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51580);