#617417

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
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

2864 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30);(5,6,7,8)(18,22)(19,25)(24,27)(26,31);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524613);