#5474046

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2199 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,27,31)(23,29,24,28)(25,36,30,26)(32,33,34,35);(7,11)(10,12)(13,14)(15,19)(16,17)(18,20)(21,23,27,24)(22,28,31,29)(25,32,30,34)(26,33,36,35);(6,9)(10,12)(14,17)(18,19)(21,24,27,23)(22,29,31,28)(25,32,30,34)(26,33,36,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,25,27,30)(22,26,31,36)(23,32,24,34)(28,33,29,35);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,18,17,19)(21,26)(22,30)(23,33)(24,35)(25,31)(27,36)(28,34)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5381242);