#10201964

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	#40139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3932 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=hi,e^2=gh,f^2=ghi,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, 10109160);