#9716162

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,5,9,15)(4,8,13,16)(10,14,11,12)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3)(2,5)(4,11)(6,9)(7,15)(8,12)(10,13)(14,16)(17,19,23,31)(18,24,28,29)(20,30,21,22)(25,27,26,32)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,7)(3,9)(8,16)(10,11)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,27,29,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,2)(3,5)(4,8)(6,7)(9,15)(10,14)(11,12)(13,16)(17,21,23,20)(18,26,28,25)(19,22,31,30)(24,32,29,27)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,4,6,13)(2,8,7,16)(3,10,9,11)(5,14,15,12)(17,22)(18,27)(19,20)(21,31)(23,30)(24,26)(25,29)(28,32)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,5)(2,3)(4,12)(6,15)(7,9)(8,11)(10,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9623358);