#348261

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×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

941 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 941 conjugacy classes of subgroups — too many to draw. Only its 168 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)(5,6,10,14)(7,15,17,20)(8,13,9,12)(11,18,19,16)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(1,2,3,4);(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,15)(14,19)(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29);(5,7)(6,11)(8,16)(9,18)(10,17)(12,15)(13,20)(14,19)(24,34)(27,32)(29,36)(30,33)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,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, 255457);