#422838

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 330034);