#561307

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1754 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 468503);