#9717704

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

3941 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,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, 9624900);