#7639806

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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)(9,10,14,11)(12,17,20,15)(13,18,21,16)(19,22,24,23)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,5)(6,8)(9,11,14,10)(12,17,20,15)(13,16,21,18)(19,22,24,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(27,34)(30,38)(33,35)(37,39);(12,20)(15,17)(19,24)(22,23)(28,36)(31,40)(33,35)(37,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^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, 7547002);