#579089

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1920 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:

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

Polycyclic presentation:

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

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, 486285);