#499014

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406210);