#9970114

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

4190 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,31)(30,32)(33,34,38,41)(35,39,44,47)(36,37,45,40)(42,43,48,46);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,27)(28,30)(31,32)(33,35,38,44)(34,39,41,47)(36,42,45,48)(37,43,40,46);(3,9)(6,13)(10,15)(14,16)(17,19,22,29)(18,23,27,24)(20,28,30,26)(21,25,31,32)(33,36,38,45)(34,40,41,37)(35,42,44,48)(39,46,47,43);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,30)(31,32)(33,36)(34,37)(35,42)(38,45)(39,43)(40,41)(44,48)(46,47);(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,20)(18,25)(19,28)(21,24)(22,30)(23,31)(26,29)(27,32)(33,35)(34,39)(36,42)(37,43)(38,44)(40,46)(41,47)(45,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,21)(18,26)(19,25)(20,23)(22,31)(24,30)(27,28)(29,32)(33,37)(34,36)(35,43)(38,40)(39,42)(41,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,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=dghi,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, 9877310);