#4308319

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215515);