#4416515

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1844 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)(3,7)(4,5)(6,8)(10,14)(11,15)(17,18)(19,20)(21,22)(23,24)(26,27)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,15,16)(17,18,21,22)(19,23,24,20);(2,4)(6,7)(10,14)(12,16)(18,22)(20,23)(25,26,29,27)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,12,14,16)(17,19)(18,23)(20,22)(21,24)(25,27,29,26)(28,31,32,30);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,14)(13,16)(17,20,21,23)(18,19,22,24)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|b^2=h,c^2=fi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323711);