#319675

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

770 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eghi,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226871);