#7634871

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

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

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, 7542067);