#7895666

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4201 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7802862);