#9947180

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	#55553	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 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)(3,4)(5,8)(6,14)(7,9)(10,15)(11,16)(12,13)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(35,41)(38,42)(43,47)(46,48);(1,3,7,5)(2,8,9,4)(6,11,15,13)(10,16,14,12)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,34,37,40)(35,42,41,38)(36,39,45,44)(43,48,47,46);(2,9)(4,8)(10,14)(12,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,4,7,8)(2,3,9,5)(6,12,15,16)(10,11,14,13)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,35,37,41)(34,38,40,42)(36,43,45,47)(39,46,44,48);(1,5,7,3)(2,8,9,4)(6,11,15,13)(10,12,14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,34)(35,38)(36,44)(37,40)(39,45)(41,42)(43,48)(46,47);(1,6,7,15)(2,10,9,14)(3,11,5,13)(4,12,8,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=g,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, 9854376);