#719831

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

2202 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,19,16)(21,22)(23,27)(24,31)(25,26)(28,33)(29,34)(30,36)(32,35);(5,7,10,9)(6,11,8,12)(13,15,17,19)(14,18,20,16)(22,26)(23,29)(28,31)(30,32);(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,16,17,18)(14,19,20,15)(21,24)(22,28)(23,30)(25,33)(26,31)(27,35)(29,32)(34,36);(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=h,d^2=fi,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, 627027);