#346894

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1981 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 1981 conjugacy classes of subgroups — too many to draw. Only its 204 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,e^c=ei,f^c=fhi

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, 254090);