#9912831

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

3947 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9820027);