#7901902

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

5194 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)(5,6)(7,8)(10,13)(14,16)(17,18)(19,27)(20,29)(21,26)(22,24)(23,28)(25,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,27)(29,31)(30,32);(2,6)(3,7)(18,24)(19,28)(25,29)(26,32);(1,3,5,7)(2,4,6,8)(17,19)(18,23)(20,26)(21,25)(22,28)(24,27)(29,31)(30,32);(1,4)(2,7)(3,6)(5,8)(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,25,24,29)(19,26,28,32)(21,27,31,23);(17,21)(18,26)(19,25)(20,23)(22,31)(24,32)(27,30)(28,29)

Polycyclic presentation:

2^9|d^2=hi,e^2=g,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, 7809098);