#7852745

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4365 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7759941);