#10480805

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	#56294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^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, 10388001);