#10336783

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	#47000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10243979);