#8360921

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

4214 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=hi,e^2=gh,f^2=h,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, 8268117);