#8930167

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

4131 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837363);