#7698966

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

5230 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)(10,14)(15,16)(17,18)(19,25)(20,29)(21,24)(22,26)(23,27)(28,32)(30,31);(9,10)(11,15)(12,16)(13,14)(17,19,24,23)(18,25,21,27)(20,30,22,28)(26,32,29,31);(2,7)(3,8)(17,20,24,22)(18,26,21,29)(19,28,23,30)(25,31,27,32);(1,3,6,8)(2,4,7,5)(17,21,24,18)(19,27,23,25)(20,26,22,29)(28,31,30,32);(1,4,6,5)(2,3,7,8)(9,11,13,12)(10,15,14,16)(17,22,24,20)(18,26,21,29)(19,28,23,30)(25,32,27,31);(1,5,6,4)(2,8,7,3)(9,12,13,11)(10,16,14,15)(17,23,24,19)(18,27,21,25)(20,30,22,28)(26,32,29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606162);