#2252990

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1747 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2160186);