#9040614

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,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, 8947810);