#9040673

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,4)(5,9)(6,14)(7,8)(10,15)(11,16)(12,13)(33,34,39,42)(35,37,46,40)(36,41,47,38)(43,45,44,48);(1,3,7,5)(2,4,8,9)(6,11,15,13)(10,12,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,35)(34,40)(36,43)(37,42)(38,45)(39,46)(41,48)(44,47);(2,8)(3,5)(6,15)(12,16)(18,22)(20,29)(23,25)(26,30)(33,36,39,47)(34,41,42,38)(35,43,46,44)(37,45,40,48);(1,4)(2,3)(5,8)(6,12)(7,9)(10,11)(13,14)(15,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,37)(34,35)(36,45)(38,44)(39,40)(41,43)(42,46)(47,48);(1,5,7,3)(2,9,8,4)(6,13,15,11)(10,16,14,12)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,36)(34,41)(35,44)(37,48)(38,42)(39,47)(40,45)(43,46);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,14)(9,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,38,39,41)(34,36,42,47)(35,45,46,48)(37,44,40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8947869);