#10480785

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	#44153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,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, 10387981);