#8359912

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	#56376	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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)(3,9,11,4)(5,8,13,14)(10,15,16,12)(17,18)(19,26)(20,21)(22,24)(23,29)(25,31)(27,28)(30,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(4,9)(8,14)(12,15)(17,19,22,29)(18,23,24,26)(20,30,31,27)(21,28,25,32)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(5,13)(8,14)(10,16)(12,15)(18,24)(21,25)(23,26)(28,32)(34,38)(39,41)(40,44)(47,48);(1,3,6,11)(2,4,7,9)(5,10,13,16)(8,12,14,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,32)(27,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4)(2,3)(5,12)(6,9)(7,11)(8,10)(13,15)(14,16)(17,20)(18,25)(19,27)(21,24)(22,31)(23,28)(26,32)(29,30)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,5,6,13)(2,8,7,14)(3,10,11,16)(4,12,9,15)(17,21,22,25)(18,20,24,31)(19,28,29,32)(23,30,26,27)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8267108);