#9947482

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18,23,27)(19,28,30,21)(20,25,32,24)(22,31,29,26)(33,34,40,42)(35,36,37,38)(39,47,48,41)(43,46,45,44);(1,2,5,8)(3,9,11,6)(4,7,13,16)(10,15,12,14)(17,19)(18,21)(20,22)(23,30)(24,31)(25,26)(27,28)(29,32)(33,35)(34,36)(37,40)(38,42)(39,43)(41,44)(45,48)(46,47);(17,20,23,32)(18,24,27,25)(19,22,30,29)(21,31,28,26)(33,36,40,38)(34,35,42,37)(39,44,48,46)(41,43,47,45);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,25)(19,29)(21,31)(22,30)(23,32)(24,27)(26,28)(33,37,40,35)(34,36,42,38)(39,45,48,43)(41,44,47,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,21)(18,19)(20,31)(22,24)(23,28)(25,29)(26,32)(27,30)(33,38,40,36)(34,35,42,37)(39,44,48,46)(41,45,47,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,22,23,29)(18,26,27,31)(19,20,30,32)(21,25,28,24)(33,39)(34,41)(35,43)(36,44)(37,45)(38,46)(40,48)(42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=i,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, 9854678);