#8091733

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4882 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7998929);