#4315106

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4222302);