#8827841

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	#41627	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,22,25)(19,23,26,20)(21,24,30,32)(27,31,29,28)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(2,6)(7,9)(8,12)(15,16)(17,19,22,26)(18,23,25,20)(21,29,30,27)(24,28,32,31)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(3,10)(7,9)(11,13)(15,16)(19,26)(20,23)(27,29)(28,31)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,29)(30,32)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)

Polycyclic presentation:

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

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, 8735037);