#8322231

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=h,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, 8229427);