#10368443

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=hi,e^2=gh,f^2=h,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, 10275639);