#104004

order	512
nilpotency class	3
isoclinism	isoclinic class of #103989
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

512 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,6)(4,7)(5,8)(10,13)(11,15)(14,16)(17,18)(19,26)(20,22)(21,31)(23,25)(24,27)(28,32)(29,30);(2,6)(3,4)(7,8)(17,19,22,28)(18,24,20,30)(21,29,23,27)(25,32,31,26);(2,7)(3,4)(6,8)(9,10,12,14)(11,16,15,13)(17,20,22,18)(19,26,28,32)(21,31,23,25)(24,27,30,29)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11200);