#8159014

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,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, 8066210);