#7879844

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4467 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=gh,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, 7787040);