#8372261

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=gi,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, 8279457);