#395620

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2249 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)(5,6,9,14,11,15,10,16)(7,12,18,27,19,29,20,28)(8,21,24,31,23,30,25,13)(17,32,33,36,22,35,34,26);(5,7)(6,12)(8,22,23,17)(9,18)(10,20)(11,19)(13,26,31,36)(14,27)(15,29)(16,28)(21,32,30,35)(24,34,25,33);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,30)(15,31)(16,21)(18,34)(19,22)(20,33)(27,35)(28,32)(29,36);(6,14)(9,10)(12,27)(13,30)(15,16)(18,20)(21,31)(24,25)(26,35)(28,29)(32,36)(33,34)

Polycyclic presentation:

2^9|a^2=f,b^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=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, 302816);