#347912

order	512
nilpotency class	3
isoclinism	isoclinic class of #347908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eh,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh,f^c=fi,g^c=gh,f^d=fh

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, 255108);