#435973

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 343169);