#9910696

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	#50792	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3882 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)(3,10)(4,9)(5,7)(6,8)(11,16)(12,15)(13,14)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(35,41)(39,45)(42,47)(46,48);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,14,15,16)(18,24)(21,25)(26,32)(29,30)(34,38)(39,45)(40,43)(46,48);(2,8)(7,10)(9,15)(14,16)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,32,30)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,21)(18,19)(20,24)(22,30)(23,25)(26,28)(27,32)(29,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,f^2=g,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, 9817892);