#406021

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1012 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=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, 313217);