#8417324

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	#52644	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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)(5,13)(7,9)(10,16)(11,12)(14,15)(18,23)(20,26)(24,30)(28,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,6)(2,8,9,4)(5,14,16,11)(10,12,13,15)(19,27)(24,30)(28,32)(29,31)(34,40)(35,42)(41,44)(43,48);(2,9)(3,6)(10,13)(11,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,3)(5,15)(6,9)(7,8)(10,14)(11,13)(12,16)(17,19,22,27)(18,24,23,30)(20,28,26,32)(21,29,25,31)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5)(2,10)(3,11)(4,12)(6,14)(7,16)(8,15)(9,13)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,31,30,29)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,3)(2,4,9,8)(5,14,16,11)(10,12,13,15)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=hi,e^2=hi,f^2=gh,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, 8324520);