#472932

order	512
nilpotency class	3
isoclinism	isoclinic class of #456658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380128);