#2227639

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2277 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2134835);