#122860

order	512
nilpotency class	4
isoclinism	isoclinic class of #122859
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	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,14,13,16,11,5,9,15,7,6,3,10,12)(17,18,20,26,21,25,33,41,22,24,34,40,36,27,44,43)(19,28,29,45,30,37,46,48,31,39,47,42,35,23,32,38);(1,3)(2,6)(4,8)(5,13)(7,15)(9,14)(10,11)(12,16)(17,19,22,31)(18,23,24,37)(20,32,34,46)(21,35,36,30)(25,39,27,28)(26,42,40,45)(29,33,47,44)(38,41,48,43);(18,24)(23,37)(25,27)(26,40)(28,39)(38,48)(41,43)(42,45)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30056);