#8227458

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

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

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, 8134654);