#4453143

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2540	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3468 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4360339);