#10247029

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

3784 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=h,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, 10154225);