#8963706

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	#47994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8870902);