#6437914

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

4533 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)(5,6,10,15,12,7,11,16)(8,19,23,17,24,20,22,13)(9,14,26,25,28,18,27,30)(21,32,35,31,36,33,34,29);(2,4)(5,7,12,6)(8,20,24,19)(9,25,28,30)(10,15,11,16)(13,23,17,22)(14,27,18,26)(21,31,36,29)(32,34,33,35);(5,8)(6,13)(7,17)(9,21)(10,22)(11,23)(12,24)(14,29)(15,20)(16,19)(18,31)(25,33)(26,34)(27,35)(28,36)(30,32);(5,9)(6,14)(7,18)(8,21)(10,26)(11,27)(12,28)(13,29)(15,25)(16,30)(17,31)(19,32)(20,33)(22,34)(23,35)(24,36);(5,10,12,11)(6,15,7,16)(8,22,24,23)(9,26,28,27)(13,20,17,19)(14,25,18,30)(21,34,36,35)(29,33,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345110);