#405988

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

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=e,e^2=i,f^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,f^c=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, 313184);