#7971831

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

4772 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,23,22)(19,27,28,24)(20,31,30,25)(21,26,29,32);(1,2,5,3)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,32)(26,31);(2,3)(6,7)(9,11,13,12)(10,14,16,15)(19,28)(21,29)(24,27)(26,32);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,16,15)(17,20,23,30)(18,25,22,31)(19,29,28,21)(24,32,27,26);(1,3,5,2)(4,6,8,7)(9,11,13,12)(10,14,16,15)(17,21,23,29)(18,26,22,32)(19,30,28,20)(24,31,27,25);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,15,16,14)(17,22,23,18)(19,24,28,27)(20,31,30,25)(21,26,29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879027);