#10412210

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	#41314	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319406);