#2606981

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2514177);