#10534316

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	#41212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(3,7)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=g,f^2=ghi,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, 10441512);