#442822

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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

957 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efgi,c^2=e,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi

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, 350018);