#712921

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2346 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:

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

Polycyclic presentation:

2^9|a^2=fgh,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, 620117);