#10408368

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

4172 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)(19,25)(20,24)(21,23)(22,29)(26,30)(27,31)(28,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(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)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(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)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(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)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(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)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gh,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, 10315564);