#346916

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2022 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)(5,6,11,15)(7,16,18,19)(8,17,12,20)(9,14,10,13)(21,22,26,24)(23,29,31,36)(25,33,35,28)(27,34,30,32);(5,7)(6,12)(8,15)(9,20)(10,17)(11,18)(13,19)(14,16)(21,23)(22,27)(24,30)(25,34)(26,31)(28,36)(29,33)(32,35);(5,8,11,12)(6,7,15,18)(9,19,10,16)(13,17,14,20)(21,24,26,22)(23,30,31,27)(25,28,35,33)(29,34,36,32);(2,4)(5,9,11,10)(6,13,15,14)(7,17,18,20)(8,19,12,16)(23,31)(27,30)(29,36)(32,34)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,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, 254112);