#7642915

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	#39486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4940 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

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, 7550111);