#10020827

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

4083 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9928023);