#8926269

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	#47053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833465);