#7971672

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×#562	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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)(3,5,7,8)(9,10,15,12)(11,16,14,18)(13,20,22,17)(19,24,21,23)(25,26)(27,30)(28,31)(29,36)(32,34)(33,37)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(25,27)(26,32)(28,30)(29,35)(31,34)(33,38)(36,39)(37,40);(10,12)(16,18)(17,20)(23,24)(26,28)(27,34)(29,37)(38,39);(9,12,15,10)(11,18,14,16)(13,20,22,17)(19,24,21,23)(25,28,31,26)(27,30,34,32)(29,36,37,33)(35,39,40,38);(9,13,15,22)(10,17,12,20)(11,19,14,21)(16,23,18,24)(25,29,31,37)(26,33,28,36)(27,35,34,40)(30,39,32,38);(1,3,4,7)(2,5,6,8)(9,14,15,11)(10,18,12,16)(13,21,22,19)(17,24,20,23)(25,30)(26,27)(28,34)(29,38)(31,32)(33,40)(35,36)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7878868);