#9711970

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	#49857	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619166);