#122865

order	512
nilpotency class	4
isoclinism	isoclinic class of #122865
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	C₂×#493	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

314 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,d^2=gi,e^2=fg,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, 30061);