#406155

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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=e,e^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,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, 313351);