#4598059

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

2263 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)(5,6,9,10)(7,8,12,11)(13,14,18,17)(15,16,20,19)(21,22)(23,27)(24,26)(25,32)(28,31)(29,33)(30,35)(34,36);(6,10)(7,12)(13,15)(14,19)(16,17)(18,20)(21,23,26,31)(22,27,24,28)(25,30,33,36)(29,34,32,35);(6,10)(8,11)(14,17)(15,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,34)(33,36);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24,26,22)(23,27,31,28)(25,32,33,29)(30,34,36,35);(5,8,9,11)(6,7,10,12)(13,17,18,14)(15,16,20,19)(21,25)(22,29)(23,30)(24,32)(26,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=fg,e^2=f,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, 4505255);