#10217027

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=hi,e^2=ghi,f^2=gi,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, 10124223);