#8341615

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4211 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8248811);