#4308425

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215621);