#9039183

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=hi,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, 8946379);