#9717603

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,27)(20,21)(22,25)(23,26)(24,30)(28,32)(29,31)(34,40)(35,38)(36,42)(45,47);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(2,7)(3,8)(9,13)(11,16)(17,20,23,22)(18,25,26,21)(19,28,30,31)(24,32,27,29)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,21,23,25)(18,20,26,22)(19,29,30,32)(24,28,27,31)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,22,23,20)(18,25,26,21)(19,28,30,31)(24,29,27,32)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,32)(26,27)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624799);