#8551017

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3827 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458213);