#556899

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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

2086 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,c^b=ch,d^b=dg,e^b=ef,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, 464095);