#411219

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265222, #72307320, #72307361
maximal normal subgroup of	#72265222, #72307320, #72307361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318415);