#626397

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(7,11)(8,12)(13,14)(15,17)(16,19)(18,20)(21,22,26,23)(24,29,31,27)(25,30,33,28)(32,34,36,35);(5,6,10,9)(7,8,11,12)(13,15,18,19)(14,17,20,16)(22,23)(24,31)(28,30)(32,36);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,35,36,34);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16,18,17)(14,19,20,15)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35);(5,9,10,6)(7,12,11,8)(13,17,18,16)(14,15,20,19)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=fhi,e^2=fg,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, 533593);