#239174

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
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

936 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)(7,15)(8,9)(11,19)(12,13)(16,20)(17,18)(21,22,27,30)(23,31,34,35)(24,25,29,26)(28,32,36,33);(5,6)(7,16)(8,12)(9,13)(10,14)(11,17)(15,20)(18,19)(21,23,27,34)(22,28,30,36)(24,35,29,31)(25,32,26,33);(2,4)(5,7)(6,11)(8,17)(9,15)(10,18)(12,19)(13,16)(14,20)(21,24,25,30,27,29,26,22)(23,28,33,31,34,36,32,35);(5,8,10,9)(6,12,14,13)(7,17,18,15)(11,19,20,16)(21,25,27,26)(22,24,30,29)(23,32,34,33)(28,35,36,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,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, 146370);