#10480796

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	#48599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10387992);