#553902

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=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, 461098);