#10168477

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	#47497	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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)(3,11,14,8)(4,15,13,9)(6,10,12,16)(18,22)(20,28)(23,25)(26,32)(34,39)(35,43)(36,44)(42,46);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48);(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,35,38,43)(34,40,39,47)(36,45,44,37)(41,42,48,46);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,44)(34,41,39,48)(35,37,43,45)(40,46,47,42);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37)(34,42)(35,36)(38,45)(39,46)(40,48)(41,47)(43,44);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,40)(36,41)(37,46)(38,39)(42,45)(43,47)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10075673);