#8934329

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	#50822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,7,11,15)(4,13,14,8)(5,12,16,9)(17,18)(19,23)(20,28)(21,25)(22,26)(24,31)(27,30)(29,32)(33,34,37,41)(35,39,43,38)(36,44,46,40)(42,47,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,18,22,26)(19,21,25,23)(20,28,31,24)(27,32,29,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(3,11)(4,14)(7,15)(8,13)(19,25)(21,23)(27,29)(30,32)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,19,22,25)(18,23,26,21)(20,29,31,27)(24,30,28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,5)(2,9)(3,8)(4,15)(6,16)(7,14)(10,12)(11,13)(17,20,22,31)(18,24,26,28)(19,27,25,29)(21,30,23,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,14)(10,15)(13,16)(17,21,22,23)(18,25,26,19)(20,30,31,32)(24,29,28,27)(33,36,37,46)(34,40,41,44)(35,42,43,45)(38,47,39,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8841525);