#8757292

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gh,e^2=i,f^2=g,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, 8664488);