#124240

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#333
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂	#7571
prime radical	C₂³	#410
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgh,d^2=gi,e^2=h,f^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31436);