#426779

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1099 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 333975);