#8260706

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(3,9)(8,13)(10,15)(17,18,23,26)(19,24,30,25)(20,31,21,22)(27,29,28,32)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,23,30)(18,24,26,25)(20,28,21,27)(22,29,31,32)(36,44)(40,46)(43,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,21)(18,22,26,31)(19,27,30,28)(24,32,25,29)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,21,23,20)(18,22,26,31)(19,28,30,27)(24,32,25,29)(33,36,39,44)(34,40,38,46)(35,43,41,47)(37,45,42,48);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,30)(24,26)(31,32)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22)(18,20)(19,29)(21,26)(23,31)(24,28)(25,27)(30,32)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=gh,e^2=ghi,f^2=ghi,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, 8167902);