#9712787

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3900 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)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(20,28)(23,30)(26,32)(29,31)(33,34,40,37)(35,43,39,41)(36,45,38,42)(44,47,46,48);(3,5)(8,10)(11,13)(15,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,31)(28,30)(29,32)(33,35,40,39)(34,41,37,43)(36,44,38,46)(42,47,45,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,40,38)(34,42,37,45)(35,44,39,46)(41,47,43,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20)(18,23)(19,26)(21,28)(22,29)(24,31)(25,30)(27,32)(33,37,40,34)(35,43,39,41)(36,42,38,45)(44,47,46,48);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,38,40,36)(34,42,37,45)(35,44,39,46)(41,48,43,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,25)(24,27)(26,29)(28,30)(31,32)(33,39,40,35)(34,43,37,41)(36,46,38,44)(42,48,45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,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, 9619983);