#8321999

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=g,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, 8229195);