#435974

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343170);