#8634133

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	#37343	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,e^2=g,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, 8541329);