#4308635

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2328 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=f,e^2=i,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, 4215831);