#8889466

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	#58335	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8796662);