#8602812

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(3,8)(4,5)(6,11)(7,9)(10,16)(12,13)(14,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,8)(4,14)(5,15)(6,9)(7,11)(10,12)(13,16)(18,22)(23,25)(24,29)(30,31)(35,42)(39,46)(43,47)(44,48);(2,9)(3,11)(5,16)(12,14)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,7,10)(2,5,9,16)(3,12,11,14)(6,15,8,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5)(2,10)(3,13)(4,9)(6,12)(7,16)(8,14)(11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,6)(2,11)(3,9)(4,13)(5,12)(7,8)(10,15)(14,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=gh,e^2=g,f^2=g,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, 8510008);