#511756

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3087 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)(6,12)(7,16)(8,19)(9,10)(13,27)(14,31)(15,26)(17,33)(18,23)(20,24)(21,22)(25,28)(29,36)(30,34);(5,6,11,15)(7,13,18,30)(8,20,22,31)(9,14,10,24)(12,21,26,19)(16,28,23,29)(17,34,33,27)(25,35,36,32);(5,7)(6,13)(8,17)(9,23)(10,16)(11,18)(12,25)(14,29)(15,30)(19,32)(20,34)(21,35)(22,33)(24,28)(26,36)(27,31);(2,4)

Polycyclic presentation:

2^9|b^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,e^b=eh,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418952);