#8295521

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	#53882	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(33,34)(35,43)(36,40)(37,41)(38,42)(39,46)(44,48)(45,47);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(33,34,38,42)(35,39,46,43)(36,40,37,41)(44,47,45,48);(2,8)(5,7)(9,12)(14,16)(18,23)(21,30)(24,31)(28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,35)(34,39)(36,44)(37,45)(38,46)(40,47)(41,48)(42,43);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47)

Polycyclic presentation:

2^9|b^2=i,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=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202717);