#2244509

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=fgh,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, 2151705);