#122587

order	512
nilpotency class	4
isoclinism	isoclinic class of #122544
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

325 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29783);