#2490668

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fgh,d^2=f,e^2=ghi,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, 2397864);