#7651235

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	#34737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4815 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558431);