#8901252

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	#47076	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,8)(4,6)(5,9)(7,11)(10,15)(12,13)(14,16)(33,34,39,44)(35,43,46,40)(36,47,48,41)(37,38,45,42);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,35,39,46)(34,40,44,43)(36,45,48,37)(38,41,42,47);(5,13)(9,12)(10,16)(14,15)(18,22)(23,29)(24,30)(31,32)(33,36,39,48)(34,41,44,47)(35,45,46,37)(38,40,42,43);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31)(33,37)(34,42)(35,36)(38,44)(39,45)(40,47)(41,43)(46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)(33,35)(34,43)(36,45)(37,48)(38,41)(39,46)(40,44)(42,47);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30)(33,38,39,42)(34,37,44,45)(35,41,46,47)(36,43,48,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808448);