#406060

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313256);