#322491

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6,10,18,8,11,9,13)(7,14,17,12,15,20,16,19)(21,22,26,28,24,29,25,30)(23,31,32,35,33,27,34,36);(1,2,4,3)(6,11)(12,19)(13,18)(14,20)(21,23,25,34,24,33,26,32)(22,27,30,35,29,31,28,36);(5,7,8,15)(6,12,11,19)(9,17,10,16)(13,20,18,14)(22,28)(23,32)(25,26)(27,31)(29,30)(33,34);(1,3,4,2)(23,33)(27,31)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,c^2=hi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229687);