#8270533

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	#40173	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4172 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,29)(27,31)(30,32)(33,34)(35,40)(36,38)(37,41)(39,43)(42,47)(44,48)(45,46);(2,6)(7,9)(8,13)(14,15)(18,23)(19,26)(20,29)(30,31)(33,35,39,47)(34,40,43,42)(36,45,37,44)(38,46,41,48);(4,12)(8,13)(10,16)(14,15)(18,23)(21,24)(25,28)(30,31)(33,36,39,37)(34,41,43,38)(35,44,47,45)(40,46,42,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,46,42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,29)(18,25,23,28)(19,27,26,32)(21,30,24,31)(33,35)(34,42)(36,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21,22,24)(18,19,23,26)(20,30,29,31)(25,27,28,32)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,45,42,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8177729);