#8772834

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	#50057	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680030);