#471654

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

938 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)(11,13)(12,14)(15,16)(17,18,20,25)(19,27,28,23)(21,31,22,26)(24,30,32,29);(1,3,5,8)(2,6,4,7)(9,11,12,15)(10,13,14,16)(17,19,22,30,20,28,21,29)(18,23,26,24,25,27,31,32);(2,4)(6,7)(17,19)(18,24)(20,28)(21,30)(22,29)(23,26)(25,32)(27,31);(1,4,5,2)(3,7,8,6)(10,14)(13,16)(18,25)(19,28)(26,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378850);