#8179655

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	#40810	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4258 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8086851);