#466984

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374180);