#8393395

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	#43192	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8300591);