#239104

order	512
nilpotency class	3
isoclinism	isoclinic class of #233055
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

910 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)(5,6,11,7)(8,16,17,20)(9,14,10,13)(12,19,15,18)(21,22,27,32)(23,24,34,36)(25,30,26,31)(28,29,35,33);(5,7,11,6)(8,16,17,20)(9,13,10,14)(12,19,15,18)(21,23,27,34)(22,28,32,35)(24,31,36,30)(25,29,26,33);(2,4)(5,8)(6,12)(7,15)(9,19)(10,18)(11,17)(13,20)(14,16)(21,24,25,35,27,36,26,28)(22,29,31,23,32,33,30,34);(8,17)(12,15)(16,20)(18,19)(21,25,27,26)(22,30,32,31)(23,29,34,33)(24,35,36,28)

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=dhi,f^c=fhi,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, 146300);