#8746456

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	#45045	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4031 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,28)(29,31)(30,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(2,7)(3,8)(9,16)(10,13)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35,38,44)(34,39,42,43)(36,41,45,46)(37,40,47,48);(2,7)(3,8)(9,16)(10,13)(19,28)(23,24)(29,32)(30,31)(33,36,38,45)(34,40,42,48)(35,41,44,46)(37,39,47,43);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36)(34,40)(35,41)(37,43)(38,45)(39,47)(42,48)(44,46);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,25)(19,29)(21,27)(22,26)(23,31)(24,30)(28,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,37,38,47)(34,41,42,46)(35,40,44,48)(36,39,45,43)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653652);