#409280

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	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,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, 316476);