#422836

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

1918 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)(7,17)(8,10)(9,23)(11,15)(12,16)(13,19)(14,26)(18,31)(20,21)(22,25)(24,29)(27,32)(28,30)(33,35)(34,36);(5,7,10,19,12,20,11,28)(6,13,15,17,16,30,8,21)(9,24,25,35,26,34,27,18)(14,31,22,36,23,33,32,29);(5,8,10,6,12,15,11,16)(7,17,28,13,20,21,19,30)(9,22,27,14,26,32,25,23)(18,33,24,29,35,31,34,36);(2,4)(5,9,12,26)(6,14,16,23)(7,18,20,35)(8,22,15,32)(10,27,11,25)(13,29,30,36)(17,33,21,31)(19,34,28,24)

Polycyclic presentation:

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

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, 330032);