#6438883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438880
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3139 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:

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

Polycyclic presentation:

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

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, 6346079);