#10166283

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

3925 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10073479);