#9924015

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 9831211);