#6443356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

5109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280597, #72283147, #72283410
maximal normal subgroup of	#72280597, #72283147, #72283410

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350552);