#722251

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1956 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)(6,7)(10,11)(21,22,27,30)(23,29,24,28)(25,35,36,26)(31,34,33,32);(14,18)(19,20)(21,23,27,24)(22,28,30,29)(25,31,36,33)(26,34,35,32);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24,27,23)(22,29,30,28)(25,31,36,33)(26,34,35,32);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,15,17,16)(14,19,18,20)(21,25)(22,26)(23,31)(24,33)(27,36)(28,34)(29,32)(30,35);(5,8,9,12)(6,10,7,11)(13,16,17,15)(14,20,18,19)(21,26)(22,25)(23,32)(24,34)(27,35)(28,33)(29,31)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=fhi,e^2=fgh,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, 629447);