#501124

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1146 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408320);