#8125484

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

4303 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)(3,4,10,8)(5,13,14,9)(11,15,16,12)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(33,34,38,41)(35,42,36,39)(37,45,46,40)(43,47,44,48);(2,7)(3,10)(9,13)(11,16)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(35,36)(39,42)(43,44)(47,48);(2,7)(4,8)(9,13)(12,15)(19,26)(21,24)(27,32)(29,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032680);