#698646

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2042 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:

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

Polycyclic presentation:

2^9|b^2=fhi,d^2=gh,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, 605842);