#562492

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

1930 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)(6,10)(8,12)(13,14,17,16)(15,18,19,20)(21,22)(23,29)(24,31)(25,33)(26,27)(28,34)(30,32)(35,36);(1,3,4,2)(6,10)(7,11)(15,19)(18,20)(22,27)(24,29)(25,34)(30,36);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(5,7)(6,8)(9,11)(10,12)(13,14,17,16)(15,18,19,20)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,20,19,18)(21,25,26,34)(22,28,27,33)(23,30,31,36)(24,32,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=f,d^2=fgh,e^2=gh,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, 469688);