#5610346

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2299 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)(16,18)(17,20)(21,22,27,24)(23,30,28,25)(26,29,35,32)(31,34,36,33);(7,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,23,27,28)(22,25,24,30)(26,31,35,36)(29,33,32,34);(6,9)(10,12)(13,14,16,18)(15,17,20,19)(21,24,27,22)(23,25,28,30)(26,32,35,29)(31,33,36,34);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,28)(23,24)(26,34)(27,30)(29,31)(32,36)(33,35);(5,7,8,11)(6,10,9,12)(13,15,16,20)(14,17,18,19)(21,26)(22,29)(23,31)(24,32)(25,33)(27,35)(28,36)(30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5517542);