#10439980

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	#48106	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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)(3,4,11,8)(5,9,14,15)(10,12,13,16)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(2,7)(4,8)(9,15)(12,16)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(1,3,6,11)(2,8,7,4)(5,13,14,10)(9,12,15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,4,6,8)(2,3,7,11)(5,12,14,16)(9,10,15,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10347176);