#239085

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
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

890 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)(5,6,11,15)(7,13,17,20)(8,18,16,12)(9,14,10,19)(21,22,27,23)(24,32,33,36)(25,30,26,29)(28,35,31,34);(5,7,11,17)(6,12,15,18)(8,9,16,10)(13,14,20,19)(21,23,27,22)(24,32,33,36)(25,29,26,30)(28,35,31,34);(2,4)(5,8,10,7,11,16,9,17)(6,13,14,18,15,20,19,12)(21,24)(22,28)(23,31)(25,35)(26,34)(27,33)(29,36)(30,32);(5,9,11,10)(6,14,15,19)(7,16,17,8)(12,20,18,13)(24,33)(28,31)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=di,f^c=fhi,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, 146281);