#362613

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1323 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269809);