#346922

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254118);