#2242168

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1772 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgh,c^2=gi,d^2=f,e^2=gh,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, 2149364);