#469630

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

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

1005 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 376826);