#8188062

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	#392	C₂³
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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,30)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,21)(18,24,28,25)(20,31,27,29)(22,30,26,32)(33,34,38,37)(35,41,43,39)(36,40,45,44)(42,47,48,46);(5,13)(8,10)(11,16)(14,15)(17,19,23,21)(18,25,28,24)(20,29,27,31)(22,30,26,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,27)(18,26,28,22)(19,29,21,31)(24,30,25,32);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,24,28,25)(20,31,27,29)(22,32,26,30)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,26)(18,27,28,20)(19,30,21,32)(24,29,25,31)(33,37,38,34)(35,39,43,41)(36,44,45,40)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=gh,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, 8095258);