#8276811

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

4234 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:

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

Polycyclic presentation:

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

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, 8184007);