#346933

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

2053 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)(5,6,11,13)(7,16,18,19)(8,17,12,20)(9,15,10,14)(21,22,27,24)(23,31,34,36)(25,30,26,29)(28,35,32,33);(5,7)(6,12)(8,13)(9,20)(10,17)(11,18)(14,19)(15,16)(21,23)(22,28)(24,32)(25,33)(26,35)(27,34)(29,31)(30,36);(5,8,11,12)(6,7,13,18)(9,19,10,16)(14,17,15,20)(21,24,27,22)(23,32,34,28)(25,29,26,30)(31,35,36,33);(2,4)(6,13)(8,12)(14,15)(16,19)(21,25,27,26)(22,29,24,30)(23,33,34,35)(28,31,32,36)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^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, 254129);