#4456263

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1996 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4363459);