#553520

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2006 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgi,d^2=fg,e^2=h,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, 460716);