#7693724

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

4818 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600920);