#9938048

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	#44659	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9845244);