#233534

order	512
nilpotency class	3
isoclinism	isoclinic class of #233521
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

1304 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 230 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)(7,15,17,19)(8,12,10,13)(11,18,20,16)(21,22,27,31)(23,32,24,35)(25,30,26,29)(28,33,36,34);(2,4)(5,7,9,17)(6,11,14,20)(8,18,10,16)(12,15,13,19)(21,23,27,24)(22,28,31,36)(25,33,26,34)(29,35,30,32);(5,6)(7,15)(8,12)(9,14)(10,13)(11,16)(17,19)(18,20)(21,24,27,23)(22,28,31,36)(25,34,26,33)(29,35,30,32);(5,8,9,10)(6,12,14,13)(7,16,17,18)(11,19,20,15)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,35,36,32)

Polycyclic presentation:

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

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, 140730);