#436475

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

1170 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)(5,6,10,15,9,16,11,14)(7,12,19,26,18,27,20,32)(8,13,23,29,22,31,24,30)(17,25,21,28,33,36,34,35);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,20)(11,19)(13,28)(14,26)(15,32)(16,27)(17,23)(22,34)(24,33)(25,29)(30,36)(31,35);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,29)(15,30)(16,31)(18,33)(19,34)(20,21)(26,35)(27,36)(28,32);(2,4)(6,14,16,15)(7,18)(12,26,27,32)(13,29,31,30)(17,33)(19,20)(21,34)(25,35,36,28)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,d^2=fi,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, 343671);