#10480867

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	#45593	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18,23,28)(19,29,21,24)(20,26,31,25)(22,30,32,27)(33,34,38,41)(35,36,43,39)(37,45,47,40)(42,48,46,44);(2,6)(7,13)(8,14)(15,16)(17,19,23,21)(18,24,28,29)(20,27,31,30)(22,26,32,25)(33,34,38,41)(35,36,43,39)(37,40,47,45)(42,44,46,48);(17,20)(18,25)(19,27)(21,30)(22,24)(23,31)(26,28)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,48)(44,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,20)(18,26)(19,30)(21,27)(22,24)(23,31)(25,28)(29,32)(33,36,38,39)(34,35,41,43)(37,44,47,48)(40,42,45,46);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21,23,19)(18,24,28,29)(20,30,31,27)(22,26,32,25)(33,35)(34,36)(37,46)(38,43)(39,41)(40,48)(42,47)(44,45);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,22,23,32)(18,27,28,30)(19,26,21,25)(20,24,31,29)(33,37)(34,40)(35,42)(36,44)(38,47)(39,48)(41,45)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=g,f^2=g,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, 10388063);