#443572

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=eg,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi

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, 350768);