#6437911

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

4627 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)(5,6,9,15,11,16,10,12)(7,17,20,28,21,27,18,13)(8,14,22,29,23,30,24,25)(19,31,33,36,34,35,32,26);(2,4)(6,12)(7,18)(8,22)(9,10)(15,16)(17,27)(20,21)(23,24)(25,29)(26,31)(32,33)(35,36);(5,7)(6,13)(8,19)(9,18)(10,20)(11,21)(12,17)(14,26)(15,27)(16,28)(22,32)(23,34)(24,33)(25,31)(29,35)(30,36);(5,8)(6,14)(7,19)(9,22)(10,24)(11,23)(12,25)(13,26)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(27,35)(28,36);(5,9,11,10)(6,15,16,12)(7,18,21,20)(8,22,23,24)(13,27,28,17)(14,29,30,25)(19,32,34,33)(26,35,36,31)

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,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, 6345107);