#249107

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1282 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1282 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156303);