#122888

order	512
nilpotency class	4
isoclinism	isoclinic class of #122865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

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

normal subgroups only

This group has 288 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,d^2=gi,e^2=fgi,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, 30084);