#147525

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi

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, 54721);