#8821941

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	#45572	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729137);