#10412336

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

4038 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)(3,8,4,5)(6,9,14,16)(11,15,12,13)(17,18)(19,25)(20,24)(21,23)(22,29)(26,30)(27,31)(28,32)(33,34,39,42)(35,40,46,48)(36,41,44,45)(37,47,43,38);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(18,21)(24,25)(26,29)(31,32)(33,35)(34,40)(36,37)(38,45)(39,46)(41,47)(42,48)(43,44);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36)(34,41)(35,43)(37,46)(38,40)(39,44)(42,45)(47,48);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32)(33,36)(34,41)(35,37)(38,48)(39,44)(40,47)(42,45)(43,46);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37)(34,38)(35,44)(36,46)(39,43)(40,41)(42,47)(45,48);(1,6,7,14)(2,9,10,16)(3,11,4,12)(5,13,8,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38)(34,37)(35,45)(36,40)(39,47)(41,46)(42,43)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,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, 10319532);