#262191

order	512
nilpotency class	3
isoclinism	isoclinic class of #246751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46000	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

938 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=hi,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169387);