#419792

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1603 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)(7,16,20,17)(8,22,25,13)(9,14)(10,27)(11,15)(12,19,29,21)(18,32)(23,28)(24,35,26,31)(30,34)(33,36);(1,2)(3,4)(5,7)(6,12)(8,23)(9,21)(10,19)(11,20)(13,30)(14,17)(15,29)(16,27)(18,24)(22,32)(25,33)(26,34)(28,31)(35,36);(2,4)(5,6)(7,17)(8,13)(9,14)(10,27)(11,15)(12,19)(16,20)(18,32)(21,29)(22,25)(23,28)(24,31)(26,35)(30,34)(33,36);(5,8,9,24,11,25,10,26)(6,13,14,31,15,22,27,35)(7,18,19,23,20,34,21,33)(12,28,16,30,29,36,17,32)

Polycyclic presentation:

2^9|a^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 326988);