#346913

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×#3567	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2081 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)(7,16,12,20)(8,18,19,17)(9,15,10,14)(21,22,26,29)(23,30,33,36)(24,28,25,34)(27,31,35,32);(5,7,11,12)(6,8,13,19)(9,18,10,17)(14,20,15,16)(21,23,26,33)(22,27,29,35)(24,32,25,31)(28,36,34,30);(5,8)(6,12)(7,13)(9,16)(10,20)(11,19)(14,18)(15,17)(21,22)(23,27)(24,28)(25,34)(26,29)(30,31)(32,36)(33,35);(2,4)(6,13)(7,12)(14,15)(17,18)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

2^9|a^2=h,b^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,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, 254109);