#4432225

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1989 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)(10,13)(11,12)(17,18)(19,22)(20,21)(23,24)(25,26,28,30)(27,29,31,32);(1,2,5,3)(4,6,8,7)(9,10,14,13)(11,16,12,15)(17,18,21,20)(19,23,24,22)(27,31)(29,32);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,23)(21,24)(25,27,28,31)(26,29,30,32);(1,3,5,2)(4,6,8,7)(9,12,14,11)(10,15,13,16)(17,19)(18,23)(20,22)(21,24)(25,26)(27,29)(28,30)(31,32);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,23,24,22)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fgi,d^2=g,e^2=f,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, 4339421);