#122597

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
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 ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 335 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,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=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29793);