#2473180

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1719 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)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,21)(20,24)(22,23)(25,26,31,32)(27,29,30,28);(9,11)(10,12)(13,16)(14,15)(18,23)(19,20)(25,27,31,30)(26,28,32,29);(10,13)(11,15)(17,19,22,20)(18,21,23,24)(25,28,31,29)(26,27,32,30);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,20,22,19)(18,21,23,24)(25,29,31,28)(26,27,32,30);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,15,12)(17,21)(18,19)(20,23)(22,24)(25,30,31,27)(26,28,32,29)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fg,d^2=f,e^2=hi,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, 2380376);