#696055

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

2058 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)(11,15)(12,16)(13,14)(17,18,21,22)(19,20,23,24)(25,26,29,32)(27,30,28,31);(2,4)(6,8)(9,11,13,16)(10,12,14,15)(17,18)(19,20)(21,22)(23,24);(3,7)(6,8)(10,14)(12,15)(18,22)(19,23)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,15)(10,11,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,28)(26,30,32,31);(1,4,5,2)(3,8,7,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,23)(19,22)(21,24)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,d^2=g,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, 603251);