#10246804

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	#50051	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,26)(20,21)(22,24)(23,30)(25,27)(28,29)(31,32);(3,9)(7,13)(10,15)(14,16)(17,19,22,30)(18,23,24,26)(20,31,27,28)(21,29,25,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(2,6)(7,13)(8,11)(14,16)(18,24)(20,27)(23,26)(28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,27)(18,25,24,21)(19,28,30,31)(23,29,26,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19)(18,26)(20,31)(21,32)(22,30)(23,24)(25,29)(27,28)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,25)(18,27,24,20)(19,29,30,32)(23,28,26,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154000);