#233459

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1324 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 274 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)(5,6,9,14)(7,15,18,12)(8,11,10,13)(16,20,17,19)(22,27)(23,31)(24,25)(29,30)(32,33)(34,35);(2,4)(6,11)(7,16)(8,10)(13,14)(17,18)(19,20)(21,22,26,30)(23,28,32,36)(24,27,25,29)(31,34,33,35);(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,20)(14,15)(21,23,26,32)(22,28,30,36)(24,31,25,33)(27,34,29,35);(5,8,9,10)(6,13,14,11)(7,16,18,17)(12,20,15,19)(21,24,26,25)(22,29,30,27)(23,31,32,33)(28,35,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140655);