#9909978

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

4227 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(18,23)(22,29)(24,27)(30,31)(34,36)(35,42)(41,44)(43,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5,6,14)(2,8,9,13)(3,11,4,12)(7,15,10,16)(17,20)(18,24)(19,25)(21,28)(22,30)(23,27)(26,32)(29,31)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)

Polycyclic presentation:

2^9|b^2=gh,d^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, 9817174);