#1147616

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2358 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=ghi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1054812);