#5277566

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2197 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22,27,30)(23,31,24,28)(25,26,36,29)(32,33,34,35);(7,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,23,27,24)(22,28,30,31)(25,34,36,32)(26,33,29,35);(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,20,19)(21,24,27,23)(22,28,30,31)(25,32,36,34)(26,33,29,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,29)(23,32)(24,34)(26,30)(27,36)(28,33)(31,35);(5,7,8,11)(6,9,10,12)(13,15,16,20)(14,17,18,19)(21,26,27,29)(22,25,30,36)(23,33,24,35)(28,34,31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=fi,e^2=fg,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, 5184762);