#4453578

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3588 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)(3,4)(5,6,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,22)(21,28)(24,27)(25,29)(26,31)(30,32);(2,3)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31);(2,3)(10,11)(14,15)(17,20,22,18)(19,23,27,24)(21,28,29,25)(26,30,32,31);(1,3,4,2)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32);(9,12,13,16)(10,14,11,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=f,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, 4360774);