#8256607

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8163803);