#636792

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4962 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)(11,15)(14,16)(18,22)(19,25)(20,28)(30,32);(10,13)(14,16)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=f,d^2=fh,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, 543988);