#2228031

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2063 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)(11,12)(13,14)(15,16)(18,19)(20,24)(26,28)(27,31);(1,3,6,5)(2,7,4,8)(9,10,13,14)(11,12,15,16)(17,18,21,19)(20,22,24,23)(27,31)(30,32);(3,5)(7,8)(10,14)(12,16)(17,18,21,19)(20,23,24,22)(25,26,29,28)(27,30,31,32);(1,4,6,2)(3,7,5,8)(9,11,13,15)(10,12,14,16)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,32)(29,31);(1,5,6,3)(2,7,4,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=i,d^2=f,e^2=fg,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, 2135227);