#7985470

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

5192 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)(3,8)(4,7)(5,6)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32);(2,6)(7,8)(9,11)(10,12)(13,16)(14,15);(1,2)(3,7)(4,8)(5,6)(9,12,14,13)(10,11,16,15)(18,23)(19,24)(25,30)(26,31);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32);(1,4,5,3)(2,7,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32);(1,2)(3,7)(4,8)(5,6)(9,13,14,12)(10,15,16,11)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)

Polycyclic presentation:

2^9|c^2=gi,d^2=gh,e^2=g,f^2=ghi,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, 7892666);