#9938219

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

4283 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)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,34)(35,43)(36,45)(37,42)(38,46)(39,41)(40,44)(47,48);(2,6)(3,9)(8,11)(10,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35,39,44)(34,40,41,43)(36,37,46,47)(38,42,45,48);(18,20)(23,25)(24,28)(30,31)(34,41)(35,44)(36,46)(42,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36)(34,38)(35,37)(39,46)(40,42)(41,45)(43,48)(44,47);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37)(34,42)(35,36)(38,40)(39,47)(41,48)(43,45)(44,46);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,38)(34,36)(35,42)(37,40)(39,45)(41,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845415);