#9814747

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

3902 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)(3,8)(4,10)(5,11)(6,13)(7,9)(12,16)(14,15)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(34,39)(35,42)(41,44)(43,48);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,14,16,11)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,47,48,45);(2,9)(3,4)(11,14)(12,13)(18,23)(22,25)(24,31)(30,32);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,11,16,14)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,45)(44,47)(46,48);(1,5)(2,11)(3,13)(4,12)(6,8)(7,15)(9,14)(10,16)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,47,40,45);(1,6,7,16)(2,12,9,13)(3,14,4,11)(5,10,15,8)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9721943);