#8183775

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	#42761	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,5)(4,12)(6,8)(7,11)(9,14)(10,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(2,8)(5,7)(9,12)(13,15)(18,22)(23,29)(24,27)(30,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,13)(14,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4,6,14)(2,9,8,12)(3,10,11,16)(5,13,7,15)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,6,7)(2,3,8,11)(4,13,14,15)(9,10,12,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=gh,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, 8090971);