#406102

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
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	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,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, 313298);