#7640820

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,20)(14,17)(15,18)(16,22)(19,23)(21,24);(2,5)(6,8)(9,11,15,17)(10,13,18,14)(12,19,22,24)(16,21,20,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,24)(18,20)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,24,20,19)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,14,15,13)(10,17,18,11)(12,21,22,23)(16,19,20,24)(27,35)(30,33)(34,40)(38,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=h,f^2=g,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, 7548016);