#10419921

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	#37527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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)(3,8)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(34,38)(36,43)(39,45)(41,44);(1,3)(2,7)(4,8)(5,10)(6,11)(9,14)(12,15)(13,16)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(34,38)(39,45)(40,46)(47,48);(1,2,6,4)(3,8,11,7)(5,9,13,12)(10,15,16,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,35,37,42)(34,39,38,45)(36,44,43,41)(40,48,46,47);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,19)(18,21)(20,26)(22,28)(23,25)(24,29)(27,30)(31,32)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,46)(44,48);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,36)(34,40)(35,41)(37,43)(38,46)(39,47)(42,44)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10327117);