#501139

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
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 ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1124 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,d^b=di,e^b=ei,f^b=fhi,d^c=di,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, 408335);