#7941708

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

4128 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=ghi,e^2=gi,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, 7848904);