#466711

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

904 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:

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

Polycyclic presentation:

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

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, 373907);