#433186

order	512
nilpotency class	3
isoclinism	isoclinic class of #419843
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,d^2=ei,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,h^b=hi,g^c=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, 340382);