#8757386

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	#55349	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664582);