#687021

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2804 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)(10,13)(11,15)(17,18,22,23)(19,21,25,31)(20,27,30,24)(26,32,28,29);(5,6,7,8)(18,23)(21,31)(24,27)(29,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,21,23,31)(20,28,30,26)(24,32,27,29);(1,2)(3,4)(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,24,23,27)(19,26,25,28)(21,29,31,32);(17,21)(18,25)(19,23)(20,29)(22,31)(24,28)(26,27)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 594217);