#560176

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3576 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 467372);