#498109

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	#2388	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

823 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405305);