#8107645

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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:

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

Polycyclic presentation:

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

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, 8014841);