#9016514

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	#47655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,15)(13,14)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(34,38)(35,43)(36,45)(42,47);(1,3,7,6)(2,8,9,16)(4,12,14,11)(5,10,15,13)(18,23)(19,26)(24,29)(27,32)(33,34,39,38)(35,40,43,48)(36,41,45,46)(37,47,44,42);(2,9)(3,6)(4,14)(5,15)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35,39,43)(34,40,38,48)(36,37,45,44)(41,47,46,42);(1,4)(2,10)(3,12)(5,16)(6,11)(7,14)(8,15)(9,13)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,45)(34,41,38,46)(35,44,43,37)(40,42,48,47);(1,5,7,15)(2,11,9,12)(3,13,6,10)(4,8,14,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,39,44)(34,42,38,47)(35,36,43,45)(40,46,48,41);(1,6,7,3)(2,8,9,16)(4,11,14,12)(5,10,15,13)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,34)(35,40,43,48)(36,46,45,41)(37,47,44,42)

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8923710);