#9716861

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	#48366	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=g,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, 9624057);