#2761354

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2901 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)(14,17)(15,19)(21,22,26,24)(23,27,30,31)(25,32,33,28)(29,35,36,34);(5,6,8,10)(7,9,12,11)(15,19)(18,20)(22,24)(27,31)(28,32)(34,35);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,30)(22,27,24,31)(25,29,33,36)(28,34,32,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,22)(23,27,30,31)(25,28,33,32)(29,35,36,34);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,25,26,33)(22,28,24,32)(23,29,30,36)(27,34,31,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2668550);