#8926720

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	#46978	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3827 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=h,e^2=gh,f^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, 8833916);