#2253890

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

1833 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)(4,6)(5,8)(9,10)(11,16)(12,13)(14,15)(17,18,20,22)(19,23,24,21)(25,26,31,29)(27,32,30,28);(1,3,6,5)(2,7,4,8)(9,10,14,15)(11,16,12,13)(19,24)(21,23)(25,27,31,30)(26,32,29,28);(1,2,6,4)(3,8,5,7)(9,11,14,12)(10,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,30)(27,29)(31,32);(1,4,6,2)(3,7,5,8)(9,12,14,11)(10,13,15,16)(17,19)(18,21)(20,24)(22,23)(25,29,31,26)(27,32,30,28);(1,5,6,3)(2,7,4,8)(9,13)(10,11)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,30,31,27)(26,32,29,28)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=fghi,d^2=h,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, 2161086);