#8351630

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

4058 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)(3,7,11,15)(4,9,14,13)(5,12,16,10)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,18,22,23)(19,24,27,31)(20,28,30,21)(25,26,29,32)(33,35,39,36)(34,37,42,40)(38,43,46,44)(41,45,48,47);(2,8)(3,11)(9,13)(10,12)(18,23)(21,28)(24,31)(26,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,4)(2,9)(3,12)(5,15)(6,14)(7,16)(8,13)(10,11)(17,19)(18,24)(20,29)(21,26)(22,27)(23,31)(25,30)(28,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,5,6,16)(2,10,8,12)(3,13,11,9)(4,7,14,15)(17,20)(18,21)(19,25)(22,30)(23,28)(24,32)(26,31)(27,29)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,21)(18,20)(19,26)(22,28)(23,30)(24,29)(25,31)(27,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8258826);