#7685218

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	#44417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4653 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)(11,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,25)(24,26)(27,30)(28,29)(31,32);(1,2,5,6)(3,7,4,8)(9,11,13,12)(10,14,16,15)(17,19,24,21)(18,22,26,20)(23,27,31,29)(25,30,32,28);(1,3,5,4)(2,7,6,8)(9,10,13,16)(11,14,12,15)(17,20,24,22)(18,19,26,21)(23,28,31,30)(25,27,32,29);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16)(17,21,24,19)(18,20,26,22)(23,29,31,27)(25,28,32,30);(9,12,13,11)(10,15,16,14)(17,22,24,20)(18,19,26,21)(23,28,31,30)(25,29,32,27);(1,4,5,3)(2,8,6,7)(17,23)(18,25)(19,27)(20,28)(21,29)(22,30)(24,31)(26,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gi,e^2=h,f^2=g,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, 7592414);