#7904710

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

5003 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,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, 7811906);