#2500007

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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=h,d^2=fgh,e^2=fhi,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, 2407203);