#404728

order	512
nilpotency class	3
isoclinism	isoclinic class of #397903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2856
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14900
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311924);