#436677

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
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	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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)(5,6,9,15,11,13,10,16)(7,17,18,26,20,12,21,28)(8,22,25,29,23,30,24,14)(19,33,35,31,34,36,32,27);(5,7)(6,12,13,17)(8,19,23,34)(9,18)(10,21)(11,20)(14,27)(15,28,16,26)(22,33)(24,32,25,35)(29,31)(30,36);(6,13)(7,18)(9,10)(12,26)(14,29)(17,28)(19,32)(20,21)(24,25)(27,33)(31,36)(34,35);(2,4)(5,8)(6,14,13,29)(7,19,20,34)(9,24)(10,25)(11,23)(12,27)(15,30,16,22)(17,31)(18,32,21,35)(26,33)(28,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,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, 343873);