#8979381

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8886577);