#9039163

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8946359);