#105248

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄²	C₂³
commutator	C₂³	C₄³
socle, prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

958 conjugacy classes of subgroups; only the 434 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 434 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=i,b^a=bd,c^a=ci,c^b=ch

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12444);