#7931602

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

4218 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)(10,12)(11,13)(17,20)(19,21)(25,26)(27,34)(28,36)(29,31)(30,33)(32,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,27,30,29)(26,31,33,34)(28,35,38,37)(32,39,36,40);(1,3,5,4)(2,7,6,8)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,29,30,27)(26,31,33,34)(28,37,38,35)(32,39,36,40);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=ghi,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, 7838798);