#9481507

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

4045 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)(3,7,10,11)(5,8,13,12)(9,14,15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,34,37,38)(35,39,43,41)(36,44,45,40)(42,47,48,46);(2,4)(7,11)(8,12)(14,16)(20,27)(24,30)(26,31)(28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(5,13)(8,12)(9,15)(14,16)(18,23)(19,25)(24,30)(26,31)(34,38)(39,41)(40,44)(46,47);(1,3,6,10)(2,7,4,11)(5,9,13,15)(8,14,12,16)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,4,6,2)(3,7,10,11)(5,12,13,8)(9,14,15,16)(17,20,22,27)(18,24,23,30)(19,26,25,31)(21,28,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5,6,13)(2,8,4,12)(3,9,10,15)(7,14,11,16)(17,21)(18,19)(20,28)(22,29)(23,25)(24,26)(27,32)(30,31)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,d^2=gi,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^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388703);