#4598999

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2015 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)(5,6,10,11)(7,12,8,9)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33);(6,11)(7,8)(13,15,16,19)(14,17,18,20)(23,30)(26,29)(31,36)(33,35);(5,7,10,8)(6,9,11,12)(13,15)(14,17)(16,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(21,23,25,30)(22,26,28,29)(24,31,32,36)(27,33,34,35);(5,9)(6,7)(8,11)(10,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4506195);