#122907

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	C₂³	#331
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 272 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) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,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, 30103);