#9923998

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

3948 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,21)(18,23,26,25)(20,27,28,29)(24,30,31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,3)(7,9)(8,10)(14,15)(17,18,22,26)(19,23,21,25)(20,24,28,31)(27,30,29,32);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,19)(18,25,26,23)(20,27,28,29)(24,30,31,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=i,e^2=gh,f^2=h,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, 9831194);