#9058581

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	#45208	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,5)(4,9)(6,7)(8,10)(11,14)(12,15)(13,16)(33,34,38,39)(35,36,43,40)(37,45,47,41)(42,48,46,44);(2,7)(3,10)(4,12)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(34,39)(36,40)(41,45)(44,48);(3,10)(4,12)(5,8)(9,15)(18,22)(19,26)(20,29)(31,32)(33,35,38,43)(34,40,39,36)(37,46,47,42)(41,44,45,48);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,36,38,40)(34,35,39,43)(37,44,47,48)(41,42,45,46);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,47)(34,41,39,45)(35,42,43,46)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^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,d^c=dgi,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, 8965777);