#9910646

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	#36795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3904 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)(3,10)(4,9)(5,7)(6,8)(11,16)(12,15)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(35,41)(39,45)(42,47)(46,48);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,14,15,16)(18,23)(24,25)(26,29)(31,32)(34,38)(39,45)(40,43)(46,48);(2,8)(7,10)(9,15)(14,16)(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);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(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);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(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);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,21,22,30)(18,26,23,29)(19,27,20,28)(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=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, 9817842);