#122915

order	512
nilpotency class	4
isoclinism	isoclinic class of #122859
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

313 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 94 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)(3,10)(4,7)(5,8)(6,11)(9,13)(12,15)(14,16)(17,18,21,25,30,23,32,28,22,26,31,24,20,19,27,29);(2,7)(3,5)(4,13)(8,15)(9,10)(11,16)(12,14)(17,19)(18,20)(21,25)(22,23)(24,31)(26,30)(27,28)(29,32);(1,3,6,5)(2,8,11,10)(4,12,14,13)(7,9,16,15)(17,20,22,30)(18,23,26,19)(21,27,31,32)(24,29,25,28)

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,c^b=ch,d^b=dg,e^b=efgh,f^b=fh,g^b=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30111);