#7917634

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4254 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)(9,10,14,16)(11,17,20,15)(12,21,22,13)(18,19,24,23)(25,26,30,34)(27,31,36,39)(28,32,35,38)(29,33,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,38)(34,39);(1,3,5,4)(2,7,6,8)(9,11,14,20)(10,15,16,17)(12,18,22,24)(13,19,21,23)(25,28,30,35)(26,32,34,38)(27,29,36,37)(31,33,39,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,23)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,13,16,21)(11,18,20,24)(15,19,17,23)(25,27)(26,31)(28,29)(30,36)(32,33)(34,39)(35,37)(38,40);(9,13)(10,12)(11,19)(14,21)(15,18)(16,22)(17,24)(20,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gh,e^2=g,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, 7824830);