#10457930

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

3976 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365126);