#8179671

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	#53060	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8086867);