#9794271

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	#58240	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,9)(4,5)(6,7)(8,12)(10,15)(11,16)(13,14)(18,21)(19,25)(24,29)(26,31)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(2,7)(4,13)(8,9)(10,16)(17,18,22,21)(19,23,25,27)(20,24,28,29)(26,30,31,32)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(17,19,22,25)(18,23,21,27)(20,26,28,31)(24,30,29,32)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,3,6,12)(2,8,7,9)(4,10,13,16)(5,11,14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,4,6,13)(2,5,7,14)(3,10,12,16)(8,11,9,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,5)(2,4)(3,11)(6,14)(7,13)(8,10)(9,16)(12,15)(17,21,22,18)(19,27,25,23)(20,24,28,29)(26,30,31,32)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 9701467);