#101873

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

366 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,6,10,13)(4,7)(5,8)(9,12,15,16)(11,14)(18,22,24,27)(19,25,26,28)(33,34,37,41,38,40,36,46)(35,42,45,47,44,48,43,39);(1,3)(2,6,8,13)(4,9)(5,10)(7,12,14,16)(11,15)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30)(33,35)(34,39,40,47)(36,43)(37,45)(38,44)(41,42,46,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16)(17,19,21,26)(18,23,24,32)(20,28,29,25)(22,30,27,31)(34,40)(36,37)(39,47)(43,45)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9069);