#8751175

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658371);