#729353

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2944 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)(10,13)(11,15)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32);(5,6,7,8)(10,13)(14,16)(18,21)(23,25)(24,28)(31,32);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31);(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29);(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fhi,d^2=fi,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, 636549);