#410848

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1269 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)(6,12)(7,16)(9,10)(13,26)(14,28)(15,24)(18,30)(19,20)(21,23)(25,29)(31,36)(33,34);(5,6,10,24,11,15,9,12)(7,17,16,13,20,32,19,26)(8,14,21,28,22,29,23,25)(18,31,30,27,33,36,34,35);(2,4)(5,7,10,16,11,20,9,19)(6,13,12,17,15,26,24,32)(8,18,21,30,22,33,23,34)(14,27,25,31,29,35,28,36);(5,8)(6,14)(7,18)(9,23)(10,21)(11,22)(12,25)(13,27)(15,29)(16,30)(17,31)(19,34)(20,33)(24,28)(26,35)(32,36)

Polycyclic presentation:

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

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, 318044);