#7966492

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5376 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)(4,7)(6,8)(9,10,14,13)(11,12,16,15)(17,18,24,29)(19,27,22,25)(20,21,30,26)(23,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11,14,16)(10,15,13,12)(17,19,24,22)(18,25,29,27)(20,31,30,23)(21,28,26,32);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,13)(14,15)(17,20,24,30)(18,26,29,21)(19,23,22,31)(25,28,27,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,21,24,26)(18,20,29,30)(19,28,22,32)(23,25,31,27);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,12,16,15)(17,22,24,19)(18,27,29,25)(20,23,30,31)(21,32,26,28);(1,2)(3,6)(4,7)(5,8)(9,12)(10,16)(11,13)(14,15)(17,23)(18,28)(19,30)(20,22)(21,25)(24,31)(26,27)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^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, 7873688);