#7630597

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	#35569	C₂
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537793);