#8366640

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	#41306	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8273836);