#9924017

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

3877 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=i,e^2=h,f^2=gh,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, 9831213);