#713264

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3192 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)(5,6,7,8)(12,16)(14,15)(17,18,22,21)(19,23,25,27)(20,24,30,29)(26,31,28,32);(2,4)(9,10,13,11)(12,14,16,15)(18,21)(19,25)(24,29)(26,28);(1,2,3,4)(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,30)(29,31);(9,12)(10,14)(11,15)(13,16)(17,20,22,30)(18,24,21,29)(19,26,25,28)(23,31,27,32);(17,21,22,18)(19,27,25,23)(20,29,30,24)(26,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fhi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 620460);