#8184837

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

3997 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,16)(13,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,37)(36,39)(38,46)(40,42)(41,48)(43,44)(45,47);(1,3,7,5)(2,8,9,11)(4,12,15,14)(6,13,10,16)(18,22)(23,25)(24,29)(30,31)(33,35,40,39)(34,37,42,36)(38,43,48,47)(41,45,46,44);(2,9)(6,10)(8,11)(13,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,40,37)(34,35,42,39)(38,44,48,45)(41,43,46,47);(1,4)(2,10)(3,12)(5,14)(6,9)(7,15)(8,16)(11,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,40,36)(34,35,42,39)(38,45,48,44)(41,43,46,47);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,16,10,13)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,38,40,48)(34,41,42,46)(35,43,39,47)(36,44,37,45);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,12,11,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,39,40,35)(34,36,42,37)(38,47,48,43)(41,44,46,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=gh,e^2=h,f^2=h,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, 8092033);