#8384925

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	#48089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=h,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, 8292121);