#8183847

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

4013 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(34,36)(35,41)(40,44)(43,45);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(18,23)(19,26)(25,28)(27,32)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,37,38,46)(34,40,36,44)(35,43,41,45)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=h,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, 8091043);