#10247626

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	#52715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=i,f^2=gh,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, 10154822);