#543430

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

1894 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 450626);