#471729

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=e,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, 378925);