#10040729

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	#52553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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)(3,8)(4,9)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34)(35,40)(36,44)(37,42)(38,46)(39,43)(41,47)(45,48);(2,7)(3,10)(4,13)(12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,40,43,46)(36,45,47,42)(37,44,48,41);(2,7)(3,10)(4,13)(12,15)(18,22)(23,25)(24,28)(30,31)(33,36,39,47)(34,41,43,44)(35,42,38,45)(37,40,48,46);(1,3,6,10)(2,8,7,14)(4,11,13,16)(5,12,9,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,42)(37,46)(38,45)(39,47)(40,48)(43,44);(1,4,6,13)(2,9,7,5)(3,11,10,16)(8,15,14,12)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37)(34,42)(35,41)(36,46)(38,44)(39,48)(40,47)(43,45);(1,5,6,9)(2,4,7,13)(3,12,10,15)(8,11,14,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,35)(34,40,43,46)(36,45,47,42)(37,41,48,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=hi,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, 9947925);