#1817494

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1979 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=gi,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1724690);