#590797

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2152 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)(5,6)(7,10)(8,12)(9,11)(14,17)(18,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(5,7,9,12)(6,10,11,8)(15,19)(18,20)(22,26)(24,33)(27,29)(30,36);(1,3,4,2)(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(5,7)(6,8)(9,12)(10,11)(13,15,16,19)(14,18,17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8)(6,7)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 497993);