#6437905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4558 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317306
maximal normal subgroup of	#72317306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345101);