#5486610

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1880 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(18,21)(22,23)(27,31)(29,32);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24)(25,26,28,30)(27,29,31,32);(2,5)(6,8)(10,14)(13,16)(17,18,20,21)(19,23,24,22)(25,26)(27,29)(28,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,22,21,23)(25,27)(26,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5393806);