#9912937

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(19,25)(22,29)(26,27)(30,31)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,28)(31,32)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,2)(3,10)(4,7)(5,13)(6,9)(8,14)(11,15)(12,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,32)(25,27)(29,31)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=h,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, 9820133);