#9814991

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	#47582	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=ghi,e^2=g,f^2=h,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, 9722187);