#8127623

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

4074 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034819);