#101250

order	512
nilpotency class	3
isoclinism	isoclinic class of #101243
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 572 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(2,5,6,7)(9,10)(11,15,16,13)(12,14)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,2,3,7,4,6,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(9,11,12,16)(10,13,14,15)(17,19,21,28)(18,24,25,32)(20,27,23,26)(22,29,30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gi,d^2=gh,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eg,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, 8446);