#7877201

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	#37714	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4541 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)(9,10,14,15)(11,12,17,20)(13,16,22,23)(18,19,24,21)(25,26)(27,36)(28,29)(30,35)(31,33)(32,38)(34,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,27,31,38)(26,32,33,36)(28,35,34,40)(29,37,39,30);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,23)(18,24)(26,33)(28,34)(32,36)(35,40);(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,34)(26,29,33,39)(27,35,38,40)(30,32,37,36);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,21)(14,20)(15,17)(16,24)(18,23)(19,22)(25,29)(26,34)(27,37)(28,33)(30,38)(31,39)(32,40)(35,36);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,16,15,23)(11,18,17,24)(12,19,20,21)(25,30,31,37)(26,35,33,40)(27,29,38,39)(28,32,34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=hi,e^2=gh,f^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, 7784397);