#10147573

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

3932 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=gi,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, 10054769);