#7966288

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

5609 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)(3,4)(5,8)(6,7)(9,10)(11,13)(12,15)(14,16)(18,23)(19,28)(20,27)(30,32);(2,7)(3,5)(9,11)(10,13)(12,16)(14,15)(17,18,22,23)(19,25,28,24)(20,26,27,21)(29,32,31,30);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,24,23,25)(20,31,27,29)(21,30,26,32);(1,4)(2,5)(3,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,32)(27,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,29)(21,23)(22,27)(24,30)(25,32)(28,31);(1,5,6,3)(2,8,7,4)(9,10)(11,14)(12,15)(13,16)(17,21,22,26)(18,27,23,20)(19,30,28,32)(24,29,25,31)

Polycyclic presentation:

2^9|b^2=h,c^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, 7873484);