#8119641

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

4164 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026837);