#548294

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1770 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)(13,14,16,18)(15,17,20,19)(21,22,26,27)(23,29,31,24)(25,33,34,28)(30,32,36,35);(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,19,20,17)(22,27)(24,29)(28,33)(32,35);(5,7,9,12)(6,10,8,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,31)(22,24,27,29)(25,30,34,36)(28,32,33,35);(1,3,4,2)(13,15)(14,17)(16,20)(18,19)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35);(5,8,9,6)(7,11,12,10)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 455490);