#7763428

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	#37616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4406 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7670624);