#416369

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(5,6,9,12,10,14,24,15)(7,16,18,27,20,11,19,25)(8,21,22,13,23,28,36,29)(17,31,32,26,34,35,33,30);(1,2,4,3)(5,7,9,19,10,20,24,18)(6,11,12,27,14,16,15,25)(8,17,22,33,23,34,36,32)(13,26,28,31,29,30,21,35);(1,3,4,2)(6,12,14,15)(11,25,16,27)(13,28,29,21)(26,35,30,31);(5,8,9,22,10,23,24,36)(6,13,15,21,14,29,12,28)(7,17,19,33,20,34,18,32)(11,26,25,35,16,30,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323565);