#8634370

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(34,39)(35,42)(36,44)(46,47);(1,3)(2,8)(4,11)(5,6)(7,10)(9,15)(12,14)(13,16)(19,20)(21,30)(24,26)(25,29)(33,34)(35,37)(36,45)(38,39)(40,44)(41,42)(43,46)(47,48);(3,10)(5,8)(11,16)(12,15)(17,19,23,20)(18,24,22,26)(21,28,30,27)(25,32,29,31)(33,35,38,42)(34,37,39,41)(36,43,44,48)(40,47,45,46);(1,4)(2,9)(3,11)(5,12)(6,14)(7,13)(8,15)(10,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36)(34,40)(35,43)(37,47)(38,44)(39,45)(41,46)(42,48);(1,5)(2,3)(4,12)(6,10)(7,8)(9,11)(13,15)(14,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,35)(34,41)(36,43)(37,39)(38,42)(40,46)(44,48)(45,47);(1,6,7,2)(3,5,10,8)(4,9,13,14)(11,15,16,12)(17,22,23,18)(19,26,20,24)(21,29,30,25)(27,32,28,31)(33,37,38,41)(34,35,39,42)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^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, 8541566);