#7850068

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

5219 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=g,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, 7757264);