#348268

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

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, 255464);