#10534369

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	#49852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,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, 10441565);