#9909880

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

4150 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 9817076);