#8395024

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

4122 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:

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

Polycyclic presentation:

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

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, 8302220);