#8741715

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	#43662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8648911);