#8137570

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	#42024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8044766);