#469443

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

1071 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)(17,18,22,27)(19,28,23,24)(20,30,21,32)(25,29,26,31);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,31,21,29)(25,30,26,32);(2,4)(6,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,22)(23,31)(24,32)(26,27)(28,30);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,20)(18,26,27,25)(19,29,23,31)(24,32,28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376639);