#9712862

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

3947 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)(3,9)(4,13)(5,7)(6,10)(8,11)(12,14)(15,16)(20,28)(23,30)(27,32)(29,31)(33,34,39,43)(35,44,46,40)(36,47,48,41)(37,38,42,45);(1,3,8,5)(2,9,11,7)(4,12,6,15)(10,16,13,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,35)(34,40)(36,45)(37,41)(38,48)(39,46)(42,47)(43,44);(1,4,8,6)(2,10,11,13)(3,12,5,15)(7,14,9,16)(17,19)(18,21)(20,27)(22,25)(23,29)(24,26)(28,32)(30,31)(33,36,39,48)(34,41,43,47)(35,45,46,38)(37,44,42,40);(1,5,8,3)(2,7,11,9)(4,12,6,15)(10,16,13,14)(17,20)(18,23)(19,27)(21,29)(22,28)(24,31)(25,32)(26,30)(33,37,39,42)(34,38,43,45)(35,41,46,47)(36,40,48,44);(1,6,8,4)(2,10,11,13)(3,12,5,15)(7,16,9,14)(17,19)(18,24)(20,27)(21,26)(22,25)(23,31)(28,32)(29,30)(33,38)(34,42)(35,36)(37,43)(39,45)(40,41)(44,47)(46,48);(1,7)(2,3)(4,14)(5,11)(6,16)(8,9)(10,12)(13,15)(17,21,22,24)(18,25,26,19)(20,29,28,31)(23,32,30,27)(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,e^2=gh,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, 9620058);