#8107294

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4071 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:

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

Polycyclic presentation:

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

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, 8014490);