#8934838

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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)(3,7)(4,5)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18,23,26)(19,24,27,32)(20,30,29,22)(21,31,28,25)(33,34)(35,37)(36,44)(38,41)(39,43)(40,47)(42,48)(45,46);(1,3,6,12)(2,7,8,9)(4,13,11,15)(5,14,10,16)(17,19)(18,24)(20,21)(22,25)(23,27)(26,32)(28,29)(30,31)(33,35,38,43)(34,39,41,37)(36,45,47,42)(40,46,44,48);(2,8)(3,12)(5,10)(13,15)(19,27)(20,29)(22,30)(24,32)(33,35,38,43)(34,37,41,39)(36,42,47,45)(40,46,44,48);(1,3)(2,9)(4,15)(5,14)(6,12)(7,8)(10,16)(11,13)(17,20)(18,22)(19,28)(21,27)(23,29)(24,31)(25,32)(26,30)(33,35)(34,37)(36,45)(38,43)(39,41)(40,48)(42,47)(44,46);(1,4,6,11)(2,10,8,5)(3,13,12,15)(7,16,9,14)(17,21,23,28)(18,25,26,31)(19,29,27,20)(22,24,30,32)(33,36,38,47)(34,40,41,44)(35,42,43,45)(37,46,39,48);(1,5,6,10)(2,11,8,4)(3,14,12,16)(7,15,9,13)(17,22)(18,20)(19,25)(21,24)(23,30)(26,29)(27,31)(28,32)(33,37)(34,35)(36,46)(38,39)(40,42)(41,43)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=gi,f^2=h,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, 8842034);