#8649851

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	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557047);