#8934308

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	#50725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,18,22,26)(19,23,27,31)(20,29,30,24)(21,28,32,25)(34,39)(36,45)(40,41)(43,47);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,34,38,39)(35,40,44,41)(36,46,45,42)(37,43,48,47);(2,5)(4,13)(7,9)(10,12)(19,27)(20,30)(23,31)(24,29)(33,35,38,44)(34,40,39,41)(36,43,45,47)(37,46,48,42);(1,4,6,13)(2,8,5,14)(3,10,11,12)(7,15,9,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,41)(36,47)(37,42)(38,44)(39,40)(43,45)(46,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,13)(14,15)(17,21)(18,25)(19,24)(20,31)(22,32)(23,30)(26,28)(27,29)(33,36,38,45)(34,42,39,46)(35,43,44,47)(37,41,48,40);(1,5,6,2)(3,9,11,7)(4,14,13,8)(10,16,12,15)(17,19)(18,23)(20,28)(21,24)(22,27)(25,30)(26,31)(29,32)(33,37,38,48)(34,43,39,47)(35,42,44,46)(36,41,45,40)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,e^2=gh,f^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,d^c=dgi,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, 8841504);