#6372029

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279225);