#7955265

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×#547	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,e^2=h,f^2=h,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, 7862461);