#8932551

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	#51749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839747);