#436655

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 343851);