#102727

order	512
nilpotency class	3
isoclinism	isoclinic class of #102709
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#2856	C₂²
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 528 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)(10,15)(11,16)(12,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,2,4,7)(3,6,8,5)(9,11,14,10)(12,16,13,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9923);