#405889

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

995 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^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, 313085);