#9053973

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	#52497	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(2,7)(4,11)(8,14)(12,16)(20,28)(23,30)(26,32)(29,31)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(3,9)(4,11)(5,13)(8,14)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,3,6,9)(2,4,7,11)(5,10,13,15)(8,12,14,16)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,46,41,48);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,16)(14,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,12,11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961169);