#4308036

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
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

2202 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)(5,6,8,10)(7,9,11,12)(13,14,18,17)(15,20,19,16)(23,29)(25,33)(27,30)(28,32);(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22,26,24)(23,30,29,27)(25,28,33,32)(31,36,35,34);(13,15)(14,16)(17,20)(18,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,19,17,15)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,7,8,11)(6,9,10,12)(13,17,18,14)(15,20,19,16)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215232);