#7701913

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dh,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, 7609109);