#7899530

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

4412 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)(11,16)(12,22)(13,15)(14,18)(17,23)(19,24)(20,21)(25,26,30,34)(27,31,36,39)(28,32,35,38)(29,33,37,40);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,14)(13,20)(15,21)(17,24)(18,22)(19,23)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,38)(34,39);(9,12,15,23)(10,17,13,22)(11,19,21,14)(16,18,20,24)(25,28,30,35)(26,32,34,38)(27,29,36,37)(31,33,39,40);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,14)(13,22)(15,23)(16,24)(18,20)(19,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,13,15,10)(11,20,21,16)(12,22,23,17)(14,18,19,24)(25,27)(26,31)(28,29)(30,36)(32,33)(34,39)(35,37)(38,40);(9,14)(10,18)(11,12)(13,24)(15,19)(16,22)(17,20)(21,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=g,e^2=gh,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, 7806726);