#2470574

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2740 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,e^2=ghi,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, 2377770);