#102518

order	512
nilpotency class	3
isoclinism	isoclinic class of #102505
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 436 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) = 512

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

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

Polycyclic presentation:

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

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, 9714);