#409790

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

900 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(18,23)(19,27)(20,29)(21,28)(24,31)(25,32);(9,11)(10,13)(12,15)(14,16)(17,18,21,26,22,23,28,30)(19,24,27,32,20,25,29,31);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,19,28,29,22,20,21,27)(18,24,30,31,23,25,26,32);(1,4,5,3)(2,6,7,8)(17,20,21,29,22,19,28,27)(18,25,26,31,23,24,30,32)

Polycyclic presentation:

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

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, 316986);