#2306172

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2350 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=fi,e^2=f,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, 2213368);