#9936480

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

4107 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)(3,8)(4,5)(6,7)(9,14)(10,11)(12,13)(15,16)(17,18)(19,26)(20,23)(21,29)(22,24)(25,30)(27,31)(28,32)(33,34,39,41)(35,40,43,37)(36,46,42,38)(44,45,47,48);(2,7)(3,9)(4,12)(11,16)(17,19,22,20)(18,23,24,26)(21,27,30,28)(25,31,29,32)(33,35,39,43)(34,40,41,37)(36,47,42,44)(38,45,46,48);(18,24)(23,26)(25,29)(31,32)(34,41)(35,43)(36,42)(45,48);(1,3,6,9)(2,8,7,14)(4,10,12,15)(5,11,13,16)(17,18)(19,26)(20,23)(21,25)(22,24)(27,32)(28,31)(29,30)(33,36,39,42)(34,38,41,46)(35,44,43,47)(37,45,40,48);(1,4,6,12)(2,5,7,13)(3,10,9,15)(8,11,14,16)(17,20,22,19)(18,23,24,26)(21,27,30,28)(25,32,29,31)(33,37)(34,35)(36,45)(38,44)(39,40)(41,43)(42,48)(46,47);(1,5)(2,4)(3,11)(6,13)(7,12)(8,10)(9,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,38)(34,42)(35,45)(36,41)(37,47)(39,46)(40,44)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=g,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, 9843676);