#4862478

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

2404 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)(5,6,9,8)(7,11,12,10)(13,14,17,19)(15,20,18,16)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36);(6,8)(10,11)(13,15,17,18)(14,16,19,20)(23,27)(24,30)(31,35)(32,36);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,18)(17,19)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,17,20)(14,18,19,15)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,7)(6,11)(8,10)(9,12)(13,14)(15,16)(17,19)(18,20)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fg,d^2=f,e^2=fg,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, 4769674);