#8260672

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	#57540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^2=ghi,f^2=hi,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, 8167868);