#4598464

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2146 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)(6,7)(8,12)(14,18)(15,20)(21,22,26,24)(23,29,30,27)(25,28,34,32)(31,36,33,35);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(22,24)(23,30)(28,32)(31,33);(8,12)(10,11)(14,18)(16,19)(21,23,26,30)(22,27,24,29)(25,33,34,31)(28,36,32,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,20,18,15)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4505660);