#8976389

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	#47088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(19,23)(20,24)(21,32)(22,26)(25,29)(27,28)(30,31)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,2,6,7)(3,8,11,16)(4,9,14,13)(5,10,15,12)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,25,30,32)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,7)(3,11)(4,14)(5,15)(17,18,22,26)(19,27,28,23)(20,30,31,24)(21,32,29,25)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,3)(2,8)(4,10)(5,13)(6,11)(7,16)(9,15)(12,14)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,4)(2,9)(3,12)(5,8)(6,14)(7,13)(10,11)(15,16)(17,20)(18,24)(19,21)(22,31)(23,32)(25,27)(26,30)(28,29)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,5)(2,10)(3,13)(4,8)(6,15)(7,12)(9,11)(14,16)(17,21)(18,25)(19,20)(22,29)(23,30)(24,27)(26,32)(28,31)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,f^2=g,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, 8883585);