#9910702

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	#2543	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,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, 9817898);