#8096841

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	#52484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004037);