#415998

order	512
nilpotency class	3
isoclinism	isoclinic class of #393999
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

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, 323194);