#422837

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

1939 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,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, 330033);