#8348940

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
prime radical	#56375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8256136);