#494542

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401738);