#8110596

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

3990 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)(3,7)(4,13)(5,15)(6,10)(8,14)(9,11)(12,16)(18,23)(19,27)(20,30)(21,31)(33,34,39,43)(35,44,41,37)(36,40,45,48)(38,47,46,42);(1,2,6,10)(3,7,12,16)(4,8,14,13)(5,9,11,15)(17,18,22,23)(19,24,27,32)(20,29,30,25)(21,26,31,28)(35,41)(36,45)(37,44)(40,48);(1,3,6,12)(2,7,10,16)(4,11,14,5)(8,15,13,9)(17,19)(18,24)(20,28)(21,29)(22,27)(23,32)(25,31)(26,30)(33,35,39,41)(34,37,43,44)(36,42,45,47)(38,40,46,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,16)(10,13)(17,20,22,30)(18,25,23,29)(19,28,27,26)(21,24,31,32)(33,36,39,45)(34,40,43,48)(35,42,41,47)(37,46,44,38);(1,2)(3,7)(4,8)(5,9)(6,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,31)(29,30)(33,37)(34,41)(35,43)(36,46)(38,45)(39,44)(40,47)(42,48);(1,5,6,11)(2,9,10,15)(3,4,12,14)(7,8,16,13)(17,21,22,31)(18,26,23,28)(19,29,27,25)(20,24,30,32)(33,38,39,46)(34,42,43,47)(35,40,41,48)(36,37,45,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8017792);