#6444414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

2654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6351610);