#346928

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
prime radical	C₂×#3553	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2037 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)(5,6,10,11)(7,13,16,18)(8,15,17,14)(9,19,20,12)(21,22,25,27)(23,29,31,36)(24,32,34,28)(26,33,35,30);(5,7)(6,8)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(5,8)(6,7)(9,13)(10,17)(11,16)(12,14)(15,19)(18,20)(21,22)(23,26)(24,32)(25,27)(28,34)(29,33)(30,36)(31,35);(2,4)(6,11)(8,17)(12,19)(13,18)(22,27)(26,35)(28,32)(29,36)

Polycyclic presentation:

2^9|a^2=h,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, 254124);