#4432227

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=gi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4339423);