#7947050

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

4085 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)(11,18)(12,22)(13,19)(14,17)(15,16)(20,21)(23,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,20)(14,21)(15,19)(16,18)(17,23)(22,24);(1,2,5,4)(3,6,8,7)(10,16)(11,19)(12,17)(21,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(9,12,15,17)(10,14,16,22)(11,20,19,23)(13,21,18,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,11)(12,21)(14,20)(15,18)(16,19)(17,24)(22,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,14)(10,17)(11,21)(12,16)(13,23)(15,22)(18,20)(19,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gi,d^2=i,e^2=gh,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, 7854246);