#346912

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×#3568	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,11)(7,13,14,17)(8,16,18,15)(9,19,20,12)(21,22,26,29)(23,30,33,36)(24,28,25,34)(27,31,35,32);(5,7)(6,8)(9,16)(10,14)(11,18)(12,17)(13,19)(15,20)(21,23)(22,27)(24,32)(25,31)(26,33)(28,36)(29,35)(30,34);(5,8)(6,7)(9,13)(10,18)(11,14)(12,15)(16,19)(17,20)(21,22)(23,27)(24,28)(25,34)(26,29)(30,31)(32,36)(33,35);(2,4)(6,11)(7,14)(12,19)(15,16)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

2^9|a^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, 254108);