#8202224

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	#47176	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109420);