#201716

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 108912);