#8926657

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	#47248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=h,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, 8833853);