#9909974

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(18,22)(20,28)(23,25)(26,32)(33,34)(35,41)(36,40)(37,46)(38,42)(39,45)(43,44)(47,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,7)(6,13)(8,11)(14,15)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,38,46)(34,37,42,40)(35,43,45,48)(39,44,41,47);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,40)(35,43)(37,42)(38,46)(39,47)(41,44)(45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,41)(36,43)(37,47)(38,45)(39,42)(40,44)(46,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,40)(34,36,42,46)(35,44,45,47)(39,43,41,48)

Polycyclic presentation:

2^9|b^2=gh,c^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, 9817170);