#10195629

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

4064 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102825);