#9711918

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	#57451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(18,23)(19,25)(26,31)(27,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,7)(4,8)(9,13)(12,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,7)(3,10)(9,13)(11,16)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,14)(12,13)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619114);