#8184750

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	#46640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=h,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, 8091946);