#403027

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1328 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=efi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310223);