#10412151

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	#41287	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)(17,18,23,25)(19,21,27,20)(22,24,31,30)(26,29,32,28)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,44)(47,48);(1,3)(2,8)(4,5)(6,13)(7,14)(9,15)(10,16)(11,12)(17,19,23,27)(18,20,25,21)(22,26,31,32)(24,28,30,29);(1,4)(2,9)(3,11)(5,14)(6,8)(7,12)(10,13)(15,16)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,4)(2,9)(3,5)(6,16)(7,12)(8,15)(10,13)(11,14)(17,18)(19,20)(21,27)(22,30)(23,25)(24,31)(26,29)(28,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(1,5)(2,6)(3,12)(4,14)(7,11)(8,9)(10,15)(13,16)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6)(2,5)(3,13)(4,8)(7,15)(9,14)(10,11)(12,16)(17,22,23,31)(18,24,25,30)(19,26,27,32)(20,28,21,29)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gi,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, 10319347);