#929863

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2090 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)(4,8)(9,10)(11,13)(12,14)(15,16)(18,19)(22,23)(26,31)(27,29);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(18,19)(22,23)(25,26,30,31)(27,28,29,32);(2,3)(6,7)(10,12)(11,16)(17,18,21,19)(20,22,24,23)(25,27,30,29)(26,32,31,28);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,15,16,13)(17,19,21,18)(20,23,24,22)(25,28,30,32)(26,27,31,29);(1,4)(2,6)(3,7)(5,8)(9,13,14,15)(10,16,12,11)(17,20,21,24)(18,22,19,23)(25,29,30,27)(26,32,31,28)

Polycyclic presentation:

2^9|b^2=f,c^2=fi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 837059);