#102997

order	512
nilpotency class	3
isoclinism	isoclinic class of #102965
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	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

679 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10193);