#10040597

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947793);