#577110

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, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fgh,e^2=fh,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, 484306);