#6437904

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

4566 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)(5,6,9,13,11,15,10,16)(7,17,18,27,21,28,20,12)(8,14,23,29,25,31,24,32)(19,30,22,35,34,36,33,26);(2,4)(5,7,11,21)(6,12,15,27)(8,22,25,33)(9,20,10,18)(13,28,16,17)(14,30,31,36)(19,24,34,23)(26,32,35,29);(6,13)(7,18)(9,10)(14,29)(15,16)(17,28)(19,22)(20,21)(23,24)(30,36)(31,32)(33,34);(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,26)(13,29)(15,31)(16,32)(17,30)(18,22)(20,33)(21,34)(27,35)(28,36);(6,15)(12,27)(13,16)(14,31)(17,28)(26,35)(29,32)(30,36)

Polycyclic presentation:

2^9|a^2=f,b^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	#72256497, #72256515, #72256521
maximal normal subgroup of	#72256497, #72256515, #72256521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345100);