#8930538

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3711 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=i,e^2=h,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, 8837734);