#8384940

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	#54947	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8292136);