#499053

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
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

1110 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,g^c=gi,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, 406249);