#422924

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

1883 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,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, 330120);