#9947066

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=i,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, 9854262);