#8922729

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

4127 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829925);