#9794281

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	#57019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 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)(3,6)(4,7)(5,8)(9,10,16,13)(11,14,17,12)(15,21,23,18)(19,20,24,22)(25,26,30,34)(27,31,36,37)(28,32,38,40)(29,33,39,35);(1,2,5,8)(3,4,7,6)(9,11)(10,14)(12,13)(15,19)(16,17)(18,22)(20,21)(23,24)(25,27,30,36)(26,31,34,37)(28,33,38,35)(29,32,39,40);(9,12,16,14)(10,17,13,11)(15,20,23,22)(18,24,21,19)(25,27)(26,31)(28,35)(29,32)(30,36)(33,38)(34,37)(39,40);(1,3)(2,4)(5,7)(6,8)(9,13,16,10)(11,12,17,14)(15,21,23,18)(19,20,24,22);(1,3)(2,6)(4,8)(5,7)(9,14,16,12)(10,17,13,11)(15,20,23,22)(18,19,21,24)(25,28)(26,32)(27,35)(29,31)(30,38)(33,36)(34,40)(37,39);(1,4,5,6)(2,7,8,3)(9,15,16,23)(10,18,13,21)(11,19,17,24)(12,20,14,22)(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=h,e^2=h,f^2=gh,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, 9701477);