#7708348

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4490 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615544);