#8898520

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	#36785	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4025 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805716);