#8119268

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	#43338	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026464);