#4308372

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2992 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=f,e^2=h,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, 4215568);