#8184877

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

4138 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=h,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, 8092073);