#9166829

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	#45912	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4033 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,25)(21,31)(22,24)(23,29)(27,32)(28,30)(35,42)(36,44)(38,39)(40,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19,22,29)(18,23,24,26)(20,27,30,31)(21,28,32,25)(33,34,37,41)(35,39,42,38)(36,40,44,45)(43,47,48,46);(3,10)(6,7)(11,16)(14,15)(18,24)(20,30)(23,26)(27,31)(33,35,37,42)(34,38,41,39)(36,43,44,48)(40,46,45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,30)(18,25,24,28)(19,27,29,31)(21,26,32,23)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,19)(18,26)(20,31)(21,25)(22,29)(23,24)(27,30)(28,32)(33,34)(35,38)(36,45)(37,41)(39,42)(40,44)(43,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21)(18,27)(19,28)(20,26)(22,32)(23,30)(24,31)(25,29)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9074025);