#5281392

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2508 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5188588);