#9712456

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18,23,27)(19,28,30,24)(20,25,31,21)(22,29,32,26)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(1,3,7,11)(2,8,9,16)(4,5,14,15)(6,12,13,10)(17,19,23,30)(18,24,27,28)(20,29,31,26)(21,32,25,22)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,9)(3,11)(4,14)(10,12)(17,20,23,31)(18,21,27,25)(19,29,30,26)(22,24,32,28)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4)(2,6)(3,5)(7,14)(8,12)(9,13)(10,16)(11,15)(17,20)(18,25)(19,29)(21,27)(22,24)(23,31)(26,30)(28,32);(1,5)(2,10)(3,4)(6,16)(7,15)(8,13)(9,12)(11,14)(17,21,23,25)(18,20,27,31)(19,22,30,32)(24,26,28,29)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,6)(2,4)(3,12)(5,8)(7,13)(9,14)(10,11)(15,16)(17,22)(18,26)(19,21)(20,24)(23,32)(25,30)(27,29)(28,31)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=hi,f^2=ghi,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, 9619652);