#9794257

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	#44783	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9701453);