#436678

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1154 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)(5,6,10,14,11,13,27,30)(7,15,25,24,17,9,18,31)(8,19,22,32,23,12,26,29)(16,21,35,36,34,33,20,28);(5,7)(6,9,13,15)(8,20)(10,25)(11,17)(12,28,19,36)(14,31,30,24)(16,22)(18,27)(21,32,33,29)(23,35)(26,34);(5,8,11,23)(6,12,13,19)(7,16,17,34)(9,21,15,33)(10,26,27,22)(14,32,30,29)(18,35,25,20)(24,36,31,28);(2,4)(5,9)(6,7,13,17)(8,21)(10,24)(11,15)(12,16,19,34)(14,18,30,25)(20,32,35,29)(22,28)(23,33)(26,36)(27,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,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, 343874);