#319777

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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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)(11,14)(12,15)(13,16)(17,21)(18,24)(19,23)(20,22)(25,26,31,38,30,28,29,33)(27,32,39,40,35,37,36,34);(1,3,4,8)(2,5,6,7)(9,11,17,18,12,19,13,20)(10,14,21,24,15,23,16,22)(25,27,31,39,30,35,29,36)(26,32,38,40,28,37,33,34);(10,15)(11,18)(13,17)(14,22)(19,20)(23,24)(25,28,30,26)(27,34,35,40)(29,33,31,38)(32,39,37,36);(2,6)(5,7)(27,35)(32,37)(34,40)(36,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=h,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, 226973);