#7641767

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	#2615	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7548963);