#7902281

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

5173 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(18,22)(19,25)(24,30)(26,27);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,16,15)(17,18,21,22)(19,23,25,29)(20,24,28,30)(26,31,27,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809477);