#440840

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1317 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=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, 348036);