#8932221

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	#47807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,10)(4,7)(5,13)(6,8)(9,14)(11,15)(12,16)(17,18,22,21)(19,25,27,23)(20,24,29,28)(26,31,32,30)(33,34)(35,41)(36,42)(37,45)(38,39)(40,46)(43,44)(47,48);(1,3,6,4)(2,7,8,10)(5,11,14,12)(9,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(34,39)(35,42)(40,45)(43,48);(2,8)(5,14)(7,10)(11,12)(17,18,22,21)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,47)(46,48);(1,2)(3,7)(4,10)(5,9)(6,8)(11,15)(12,16)(13,14)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,6,3)(2,7,8,10)(5,12,14,11)(9,15,13,16)(17,21,22,18)(19,23,27,25)(20,24,29,28)(26,31,32,30)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,5,6,14)(2,9,8,13)(3,11,4,12)(7,15,10,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839417);