#7697482

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

4837 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604678);