#626502

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=f,d^2=fhi,e^2=g,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, 533698);