#7903124

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	#50640	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4444 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7810320);