#8184883

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

4199 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 8092079);