#10504606

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^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, 10411802);