#8389282

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

4098 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,e^2=g,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, 8296478);