#8418148

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	#45257	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(18,24)(19,21)(26,31)(27,29)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(3,12)(6,13)(8,16)(10,15)(17,19,23,21)(18,25,24,20)(22,29,30,27)(26,28,31,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,21,23,19)(18,25,24,20)(22,29,30,27)(26,32,31,28)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)(17,22,23,30)(18,26,24,31)(19,27,21,29)(20,28,25,32)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

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, 8325344);