#419893

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327089);