#580982

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

1766 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)(8,11)(21,22)(23,25)(24,32)(26,27)(28,30)(29,33)(31,35)(34,36);(1,2,4,3)(6,10)(7,12)(14,17)(18,20)(22,27)(23,30)(24,33)(34,35);(5,6)(7,11)(8,12)(9,10)(13,14,16,17)(15,18,19,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,34)(33,36);(1,3,4,2)(5,7,9,12)(6,11,10,8)(21,24)(22,29)(23,31)(25,34)(26,33)(27,32)(28,35)(30,36);(5,8,9,11)(6,12,10,7)(13,15,16,19)(14,18,17,20)(21,25,26,28)(22,30,27,23)(24,34,33,35)(29,36,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=fgi,e^2=h,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, 488178);