#2242187

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2276 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=fgi,d^2=f,e^2=fgh,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, 2149383);