#416258

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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:

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

Polycyclic presentation:

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

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, 323454);