#8417048

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

3979 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(17,18,23,27)(19,25,29,22)(20,24,21,26)(28,30,31,32)(33,34,39,40)(35,44,45,41)(36,48,42,37)(38,43,46,47);(2,7)(4,13)(8,16)(10,14)(17,19,23,29)(18,22,27,25)(20,28,21,31)(24,32,26,30)(34,40)(35,45)(36,42)(43,47);(2,7)(5,15)(8,16)(9,12)(17,20,23,21)(18,24,27,26)(19,28,29,31)(22,32,25,30)(33,35)(34,41)(36,46)(37,47)(38,42)(39,45)(40,44)(43,48);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,19)(18,25)(20,28)(21,31)(22,27)(23,29)(24,30)(26,32)(33,36,39,42)(34,37,40,48)(35,46,45,38)(41,47,44,43);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,21,23,20)(18,26,27,24)(19,28,29,31)(22,32,25,30)(33,37)(34,42)(35,47)(36,40)(38,41)(39,48)(43,45)(44,46);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,22,23,25)(18,19,27,29)(20,30,21,32)(24,31,26,28)(33,38,39,46)(34,43,40,47)(35,42,45,36)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324244);