#8934859

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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=gh,e^2=gi,f^2=h,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, 8842055);