#6439026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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

3064 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6346222);