#9783084

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
prime radical	#56607	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,41)(35,40,43,38)(36,37,42,48)(44,45,47,46);(2,6)(7,9)(8,12)(14,15)(20,27)(24,30)(26,31)(28,32)(33,35)(34,40)(36,44)(37,45)(38,41)(39,43)(42,47)(46,48);(18,23)(19,25)(24,30)(26,31)(34,41)(35,43)(36,42)(45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,42)(34,37,41,48)(35,44,43,47)(38,46,40,45);(1,2)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,42)(35,45)(36,41)(38,44)(39,48)(40,47)(43,46);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,43)(35,41)(36,46)(37,47)(39,40)(42,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9690280);