#7850846

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

4709 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7758042);