#4866700

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 2097152
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

1968 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4773896);