#8908326

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	#51291	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3972 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815522);