#7699462

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

4715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=gh,e^2=gi,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, 7606658);