#9717610

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	#57175	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(19,28)(20,30)(21,25)(22,29)(23,24)(26,31)(27,32)(33,34)(35,44)(36,42)(37,43)(38,41)(39,40)(45,47)(46,48);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(18,24)(19,21)(26,30)(27,29)(34,40)(37,47)(41,44)(42,48);(2,7)(3,8)(9,13)(11,16)(17,19,23,21)(18,25,24,28)(20,27,31,29)(22,26,32,30)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,20,23,31)(18,26,24,30)(19,29,21,27)(22,28,32,25)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,21,23,19)(18,25,24,28)(20,27,31,29)(22,30,32,26)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,22)(18,27)(19,26)(20,28)(21,30)(23,32)(24,29)(25,31)(33,38,39,35)(34,44,40,41)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=ghi,f^2=hi,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, 9624806);