#417580

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2380 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:

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

Polycyclic presentation:

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

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, 324776);