#7961770

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	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=gi,e^2=g,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, 7868966);