#9712857

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3826 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,f^2=i,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, 9620053);