#1672356

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1910 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,d^2=f,e^2=gh,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, 1579552);