#8188571

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(18,20)(19,25)(24,28)(27,31)(33,34,39,41)(35,40,37,47)(36,44,46,38)(42,43,45,48);(2,3)(7,9)(8,10)(14,15)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,35,39,37)(34,40,41,47)(36,45,46,42)(38,43,44,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,36,39,46)(34,38,41,44)(35,42,37,45)(40,48,47,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,37,39,35)(34,40,41,47)(36,45,46,42)(38,48,44,43);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38)(34,36)(35,43)(37,48)(39,44)(40,45)(41,46)(42,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=ghi,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, 8095767);