#122909

order	512
nilpotency class	4
isoclinism	isoclinic class of #122865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₈²	C₂³
prime radical	#3147	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

282 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gi,e^2=fgh,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30105);