#8321990

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(4,10)(5,6)(7,9)(11,13)(12,15)(14,16)(33,34,40,44)(35,43,38,39)(36,42,37,41)(45,48,46,47);(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)(33,35,40,38)(34,39,44,43)(36,45,37,46)(41,48,42,47);(2,9)(3,6)(10,16)(11,12)(18,23)(21,31)(24,29)(25,32)(33,36,40,37)(34,41,44,42)(35,45,38,46)(39,48,43,47);(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)(33,37,40,36)(34,42,44,41)(35,45,38,46)(39,48,43,47);(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)(33,38,40,35)(34,43,44,39)(36,46,37,45)(41,47,42,48);(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)(33,39,40,43)(34,35,44,38)(36,47,37,48)(41,46,42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,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, 8229186);