#9712053

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3942 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=gh,e^2=g,f^2=i,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, 9619249);