#8192584

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

4161 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)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34,38,40)(35,39,42,36)(37,45,46,41)(43,44,48,47);(1,2,5,6)(3,7,10,12)(4,8,11,13)(9,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35)(34,39)(36,40)(37,43)(38,42)(41,47)(44,45)(46,48);(4,11)(8,13)(9,15)(14,16)(18,21)(23,25)(24,28)(30,31)(34,40)(35,42)(41,45)(43,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,36,38,39)(34,35,40,42)(37,44,46,47)(41,43,45,48);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,35)(34,39)(36,40)(37,43)(38,42)(41,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,e^2=ghi,f^2=g,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, 8099780);