#7674167

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	#45417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4518 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)(12,16)(13,15)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,14,15,12)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32);(9,12)(10,11)(13,14)(15,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,25,28,32);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,16)(13,15)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25);(1,3,5,4)(2,7,6,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31);(1,4,5,3)(2,8,6,7)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581363);