#7883114

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(9,10,14,11)(12,17,20,15)(13,16,22,18)(19,24,21,23)(25,26,30,29)(27,31,33,35)(28,32,36,37)(34,38,39,40);(1,3,4,8)(2,5,6,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(27,33)(31,35)(34,39)(38,40);(10,11)(15,17)(16,18)(23,24)(25,27,30,33)(26,31,29,35)(28,34,36,39)(32,38,37,40);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,26)(27,35,33,31)(28,32,36,37)(34,38,39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=i,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, 7790310);