#101233

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 470 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

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, 8429);