#416284

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

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, 323480);