#346370

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2238 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 2238 conjugacy classes of subgroups — too many to draw. Only its 232 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253566);