#636186

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 543382);