#10368462

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275658);