#10247483

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	#52336	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,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, 10154679);