#9938220

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9845416);