#10147807

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

4276 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(2,7)(4,15)(8,11)(10,13)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(35,41)(38,45)(42,47)(46,48);(2,7)(3,12)(9,14)(10,13)(18,25)(19,30)(20,27)(28,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=g,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, 10055003);