#10216912

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=i,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10124108);