#7699463

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
prime radical	#59818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=gh,e^2=ghi,f^2=i,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, 7606659);