#319252

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

744 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)(9,10,14,16,13,18,12,17)(11,19,21,15,20,24,22,23)(25,26,30,34,29,33,36,39)(27,35,37,32,28,31,38,40);(1,3,4,8)(2,5,6,7)(11,20)(15,23)(19,24)(21,22)(25,27,36,38,29,28,30,37)(26,31,39,32,33,35,34,40);(9,11,13,20)(10,15,18,23)(12,21,14,22)(16,19,17,24)(25,28,29,27)(26,32,33,40)(30,37,36,38)(31,34,35,39);(2,6)(5,7)(9,12,13,14)(10,16,18,17)(11,21,20,22)(15,19,23,24)

Polycyclic presentation:

2^9|a^2=fhi,b^2=eghi,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 226448);