#419920

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1612 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,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, 327116);