#411337

order	512
nilpotency class	3
isoclinism	isoclinic class of #403928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2615
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15113
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1108 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)(4,17)(5,12)(6,10)(7,11)(8,15)(9,18)(14,28)(16,24)(19,26)(20,23)(21,27)(22,31)(25,30)(29,32);(1,3,6,16,7,15,5,20)(2,8,12,24,11,13,10,23)(4,14,18,31,19,29,21,30)(9,25,17,32,27,22,26,28);(1,4)(2,9)(3,14)(5,21)(6,18)(7,19)(8,22)(10,26)(11,27)(12,17)(13,25)(15,29)(16,31)(20,30)(23,32)(24,28);(2,10,11,12)(8,23,13,24)(9,26,27,17)(22,32,25,28)

Polycyclic presentation:

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

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, 318533);