#347905

order	512
nilpotency class	3
isoclinism	isoclinic class of #347878
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1025 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)(3,5)(4,6)(7,8)(9,10,15,18)(11,14,21,30)(12,22,26,19)(13,17,29,34)(16,24,32,25)(20,27,33,28)(23,35,38,36)(31,39,40,37);(2,5)(6,7)(9,11,14,18,15,21,30,10)(12,19,24,32,26,22,25,16)(13,20,27,34,29,33,28,17)(23,35,37,40,38,36,39,31);(2,6)(4,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,31)(18,32)(20,35)(21,22)(25,30)(27,37)(28,39)(29,38)(33,36)(34,40);(9,13)(10,17)(11,20)(12,23)(14,27)(15,29)(16,31)(18,34)(19,35)(21,33)(22,36)(24,37)(25,39)(26,38)(28,30)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,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, 255101);