#696697

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

2120 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,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, 603893);