#8402348

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

3989 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,8)(5,7)(9,12)(14,16)(18,23)(21,25)(24,26)(28,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=ghi,e^2=h,f^2=gi,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, 8309544);