#712920

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2604 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=fg,e^2=fg,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, 620116);