#737663

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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

3904 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,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, 644859);