#8425520

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332716);