#8136944

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

4085 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044140);