#369107

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35329	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1900 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276303);