#9910820

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3820 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,15)(12,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(35,41)(39,45)(42,47)(46,48);(1,3,7,11)(2,8,9,15)(4,12,14,5)(6,10,13,16)(18,23)(24,25)(26,29)(31,32)(34,38)(39,45)(40,43)(46,48);(2,9)(3,11)(4,14)(10,16)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,13)(15,16)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5,7,12)(2,10,9,16)(3,4,11,14)(6,8,13,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,6)(2,4)(3,10)(5,15)(7,13)(8,12)(9,14)(11,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818016);