#9910135

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,e^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, 9817331);