#346915

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×#3559	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2214 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)(7,16,12,20)(8,18,19,17)(9,14,10,13)(21,22,25,28)(23,29,30,36)(24,32,34,27)(26,33,35,31);(5,7,11,12)(6,8,15,19)(9,18,10,17)(13,20,14,16)(21,23,25,30)(22,26,28,35)(24,33,34,31)(27,36,32,29);(5,8)(6,12)(7,15)(9,16)(10,20)(11,19)(13,18)(14,17)(24,34)(27,32)(29,36)(31,33);(2,4)(5,9,11,10)(6,13,15,14)(7,17,12,18)(8,16,19,20)(23,30)(26,35)(29,36)(31,33)

Polycyclic presentation:

2^9|a^2=h,b^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, 254111);