#9947459

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9854655);