#8393628

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	#49624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(1,3,7,4)(2,8,9,10)(5,14,16,12)(6,13,11,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,34)(35,40)(36,42)(37,39)(38,41)(43,47)(44,48)(45,46);(2,9)(6,11)(8,10)(13,15)(18,23)(21,29)(24,25)(27,32)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,4,7,3)(2,10,9,8)(5,12,16,14)(6,15,11,13)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5)(2,11)(3,12)(4,14)(6,9)(7,16)(8,13)(10,15)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,6,7,11)(2,5,9,16)(3,13,4,15)(8,14,10,12)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8300824);