#7839890

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7747086);