#9910803

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=gh,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, 9817999);