#8322181

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,5)(4,9)(6,11)(7,8)(10,15)(12,14)(13,16)(33,34)(35,44)(36,38)(37,46)(39,43)(40,47)(41,48)(42,45);(1,3)(2,5)(4,6)(7,10)(8,15)(9,11)(12,13)(14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(2,8)(3,10)(6,12)(9,16)(18,23)(21,31)(24,29)(25,32)(33,36,39,48)(34,41,43,38)(35,45,47,37)(40,42,44,46);(1,4)(2,9)(3,12)(5,14)(6,10)(7,13)(8,16)(11,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,29)(27,30)(31,32)(33,37)(34,42)(35,36)(38,40)(39,45)(41,44)(43,46)(47,48);(1,5,7,15)(2,10,8,3)(4,11,13,14)(6,9,12,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,32)(24,31)(26,28)(33,35)(34,40)(36,45)(37,48)(38,46)(39,47)(41,42)(43,44);(1,6,7,12)(2,11,8,14)(3,4,10,13)(5,9,15,16)(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,29,30,24)(33,38,39,41)(34,36,43,48)(35,46,47,42)(37,40,45,44)

Polycyclic presentation:

2^9|c^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8229377);