#755519

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2044 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=gh,d^2=i,e^2=i,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, 662715);