#7919110

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

4956 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,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, 7826306);