#9938073

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	#41313	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,f^2=gh,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, 9845269);