#8159185

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

4053 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066381);