#240519

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
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	C₂×#3247	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

969 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:

(6,11)(7,15)(8,9)(12,19)(13,14)(16,20)(21,22,26,30)(23,27,32,36)(24,29,25,28)(31,34,33,35);(1,2)(3,4)(5,6)(7,16)(8,14)(9,11)(10,13)(12,18)(15,19)(17,20)(21,23)(22,27)(24,33)(25,31)(26,32)(28,34)(29,35)(30,36);(2,4)(5,7,8,17,10,15,9,18)(6,12,11,19,13,20,14,16)(22,28)(24,25)(27,34)(29,30)(31,33)(35,36);(6,13)(11,14)(12,20)(16,19)(21,24,26,25)(22,29,30,28)(23,31,32,33)(27,34,36,35)

Polycyclic presentation:

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

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, 147715);