#8933189

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	#48114	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840385);