#8183724

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

4219 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090920);