#9716170

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9623366);