#8156959

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	#53420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 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)(3,9)(4,5)(6,14)(7,8)(10,15)(11,12)(13,16)(18,23)(19,27)(21,25)(28,30)(36,44)(39,46)(42,48)(45,47);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,29)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,8)(4,9)(10,14)(12,16)(19,27)(24,31)(28,30)(29,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,19,22,27)(18,24,23,31)(20,30,26,28)(21,29,25,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20,22,26)(18,25,23,21)(19,28,27,30)(24,29,31,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,6,7,15)(2,10,8,14)(3,11,5,13)(4,12,9,16)(17,21,22,25)(18,26,23,20)(19,29,27,32)(24,28,31,30)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064155);