#239154

order	512
nilpotency class	3
isoclinism	isoclinic class of #233301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146350);