#8932773

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	#47603	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839969);