#8253885

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

4228 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,25,26,21)(20,28,29,24)(27,30,32,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(2,6)(7,9)(8,13)(14,15)(18,23)(19,26)(20,29)(30,31)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(4,12)(8,13)(10,16)(14,15)(18,23)(21,25)(24,28)(30,31)(34,38)(39,41)(40,44)(46,47);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,31)(29,32)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,29)(18,24,23,28)(19,27,26,32)(21,30,25,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,21)(18,19)(20,30)(22,25)(23,26)(24,27)(28,32)(29,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

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

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, 8161081);