#10424696

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=g,f^2=h,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, 10331892);