#8197297

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	#48642	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104493);