#9039166

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3780 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946362);